diff options
33 files changed, 639 insertions, 145 deletions
@@ -1,4 +1,3 @@ - blog/rss.org sitemap.org sitemap.xml @@ -73,13 +73,23 @@ DEADLINE: <2026-03-18 Wed> These are some habits I want to track. They are repeated according to a calendar schedule in general. ** TODO Merge Feature Branches to Monorepo Main -SCHEDULED: <2026-03-29 Sun 10:00 .+1w> +SCHEDULED: <2026-04-05 Sun 10:00 ++1w> +:PROPERTIES: +:LAST_REPEAT: [2026-04-01 Wed 13:48] +:END: +- State "DONE" from "TODO" [2026-04-01 Wed 13:48] I need to have a schedule for cleaning my git tree. ** TODO Skincare -SCHEDULED: <2026-03-30 Mon 12:30 .+1d> +SCHEDULED: <2026-04-12 Sun 12:30 .+1d> :PROPERTIES: -:LAST_REPEAT: [2026-03-29 Sun 14:01] +:LAST_REPEAT: [2026-04-11 Sat 13:20] :END: +- State "DONE" from "TODO" [2026-04-11 Sat 13:20] +- State "DONE" from "TODO" [2026-04-09 Thu 20:01] +- State "DONE" from "TODO" [2026-04-06 Mon 20:45] +- State "DONE" from "TODO" [2026-04-05 Sun 16:59] +- State "DONE" from "TODO" [2026-04-04 Sat 16:36] +- State "DONE" from "TODO" [2026-03-30 Mon 14:01] - State "DONE" from "TODO" [2026-03-29 Sun 14:01] - State "DONE" from "TODO" [2026-03-27 Fri 11:29] - State "DONE" from "TODO" [2026-03-19 Thu 03:36] @@ -90,11 +100,20 @@ Current stack: - Hyaluronic acid (2x/day) - Glycerin for hands ** TODO Supplements -SCHEDULED: <2026-03-30 Mon 12:00 .+1d> +SCHEDULED: <2026-04-12 Sun 12:00 .+1d> :PROPERTIES: -:LAST_REPEAT: [2026-03-29 Sun 14:01] +:LAST_REPEAT: [2026-04-11 Sat 13:20] :STYLE: habit :END: +- State "DONE" from "TODO" [2026-04-11 Sat 13:20] +- State "DONE" from "TODO" [2026-04-09 Thu 20:01] +- State "DONE" from "TODO" [2026-04-07 Tue 14:40] +- State "DONE" from "TODO" [2026-04-06 Mon 20:44] +- State "DONE" from "TODO" [2026-04-05 Sun 16:59] +- State "DONE" from "TODO" [2026-04-04 Sat 16:36] +- State "DONE" from "TODO" [2026-04-02 Thu 22:09] +- State "DONE" from "TODO" [2026-04-01 Wed 13:48] +- State "DONE" from "TODO" [2026-03-30 Mon 14:01] - State "DONE" from "TODO" [2026-03-29 Sun 14:01] - State "DONE" from "TODO" [2026-03-28 Sat 17:44] - State "DONE" from "TODO" [2026-03-27 Fri 11:29] @@ -160,11 +179,12 @@ SCHEDULED: <2025-09-29 Mon 16:00 .+1d> - State "DONE" from "TODO" [2025-06-26 Thu 23:22] I want to be able to run or bike every day so that I get my exercise in. ** TODO Stretch -SCHEDULED: <2026-03-20 Fri 00:00 .+1d> +SCHEDULED: <2026-04-04 Sat 00:00 .+1d> :PROPERTIES: -:LAST_REPEAT: [2026-03-19 Thu 03:36] +:LAST_REPEAT: [2026-04-03 Fri 00:01] :STYLE: habit :END: +- State "DONE" from "TODO" [2026-04-03 Fri 00:01] - State "DONE" from "TODO" [2026-03-19 Thu 03:36] - State "DONE" from "TODO" [2025-09-24 Wed 07:08] - State "DONE" from "TODO" [2025-09-21 Sun 06:57] @@ -185,11 +205,14 @@ Stretches: - Seal - Cat ** TODO Journal -SCHEDULED: <2026-03-31 Tue 22:00 .+1d> +SCHEDULED: <2026-04-12 Sun 22:00 .+1d> :PROPERTIES: -:LAST_REPEAT: [2026-03-30 Mon 04:00] +:LAST_REPEAT: [2026-04-11 Sat 13:21] :STYLE: habit :END: +- State "DONE" from "TODO" [2026-04-11 Sat 13:21] +- State "DONE" from "TODO" [2026-04-02 Thu 22:09] +- State "DONE" from "TODO" [2026-04-01 Wed 00:22] - State "DONE" from "TODO" [2026-03-30 Mon 04:00] - State "DONE" from "TODO" [2026-03-28 Sat 22:21] - State "DONE" from "TODO" [2026-03-27 Fri 00:00] @@ -222,10 +245,18 @@ SCHEDULED: <2026-03-31 Tue 22:00 .+1d> - State "DONE" from "TODO" [2025-01-11 Sat 02:25] I want to journal every day, at least a little bit, about my life and track it with a git repo. ** TODO Update Agenda -SCHEDULED: <2026-03-31 Tue 11:00 .+1d> +SCHEDULED: <2026-04-12 Sun 11:00 .+1d> :PROPERTIES: -:LAST_REPEAT: [2026-03-30 Mon 17:08] +:LAST_REPEAT: [2026-04-11 Sat 13:20] :END: +- State "DONE" from "TODO" [2026-04-11 Sat 13:20] +- State "DONE" from "TODO" [2026-04-09 Thu 20:01] +- State "DONE" from "TODO" [2026-04-07 Tue 14:40] +- State "DONE" from "TODO" [2026-04-06 Mon 20:44] +- State "DONE" from "TODO" [2026-04-05 Sun 16:59] +- State "DONE" from "TODO" [2026-04-04 Sat 16:36] +- State "DONE" from "TODO" [2026-04-02 Thu 22:09] +- State "DONE" from "TODO" [2026-04-01 Wed 13:48] - State "DONE" from "TODO" [2026-03-30 Mon 17:08] - State "DONE" from "TODO" [2026-03-29 Sun 14:01] - State "DONE" from "TODO" [2026-03-28 Sat 17:44] diff --git a/config/emacs.org b/config/emacs.org index 6dc0719..375c31e 100644 --- a/config/emacs.org +++ b/config/emacs.org @@ -327,6 +327,25 @@ then append the typed input to the mu4e database query." (if (facep face) (funcall orig-fn face attribute frame inherit) 'unspecified)) + +(defun rp/vterm-cleanup-on-exit (buffer _event) + "Close windows showing BUFFER after vterm exits, then kill BUFFER." + (let ((buf buffer)) + (run-at-time + 0 nil + (lambda () + (when (buffer-live-p buf) + (let ((wins (get-buffer-window-list buf nil t))) + (dolist (win wins) + (when (window-live-p win) + (if (one-window-p t win) + ;; Can't delete the last window in a frame. + ;; Switch it away from the vterm buffer instead. + (with-selected-window win + (switch-to-prev-buffer win 'kill)) + (delete-window win))))) + (when (buffer-live-p buf) + (kill-buffer buf))))))) #+end_src ** Random Packages These are packages that I require in order to write some scripts in emacs-lisp. @@ -394,18 +413,13 @@ Emacs is self documenting, after all! (display-line-numbers-type 'relative "Relative line numbers for easy vim jumping") (use-short-answers t "Use y instead of yes") (make-backup-files nil "Don't make backups") - (display-fill-column-indicator-column 150 "Draw a line at 100 characters") - (fill-column 150) (line-spacing 2 "Default line spacing") (c-doc-comment-style '((c-mode . doxygen) (c++-mode . doxygen))) + (fill-column 150) - - :hook ((text-mode . visual-line-mode) - (prog-mode . display-line-numbers-mode) - (prog-mode . display-fill-column-indicator-mode) + :hook ((prog-mode . display-line-numbers-mode) (org-mode . auto-fill-mode) - (org-mode . display-fill-column-indicator-mode) (org-mode . display-line-numbers-mode)) :config (emacs-config)) #+end_src @@ -443,6 +457,8 @@ This is my org mode configuration, which also configures latex. (org-edit-src-content-indentation 0) (org-src-tab-acts-natively t) (org-src-preserve-indentation t) + (org-hide-drawer-startup t) + (org-startup-folded 'showall) (TeX-PDF-mode t) (org-confirm-babel-evaluate nil "Don't ask to evaluate code block") @@ -469,6 +485,7 @@ This is my org mode configuration, which also configures latex. \\setlength{\\topmargin}{1.5cm} \ \\addtolength{\\topmargin}{-2.54cm} \ \\usepackage{amsmath} \ + \\usepackage{tikz-cd} \ ") (org-preview-latex-image-directory (expand-file-name "~/.cache/ltximg/") "don't use weird cache location") (org-latex-preview-ltxpng-directory (expand-file-name "~/.cache/ltximg/") "don't use weird cache location") @@ -478,7 +495,7 @@ This is my org mode configuration, which also configures latex. (TeX-engine 'xetex "set xelatex as default engine") (preview-default-option-list '("displaymath" "textmath" "graphics") "preview latex") ;; (preview-image-type 'png "Use PNGs") - (org-preview-latex-default-process 'dvipng) + ;; (org-preview-latex-default-process 'imagemagick) (org-format-latex-options '(:foreground default :background default @@ -490,7 +507,6 @@ This is my org mode configuration, which also configures latex. (org-return-follows-link t "be able to follow links without mouse") (org-startup-indented t "Indent the headings") (org-image-actual-width '(300) "Cap width") - (org-startup-with-latex-preview t "see latex previews on opening file") (org-startup-with-inline-images t "See images on opening file") (org-hide-emphasis-markers t "prettify org mode") (org-use-sub-superscripts "{}" "Only display superscripts and subscripts when enclosed in {}") @@ -508,7 +524,6 @@ This is my org mode configuration, which also configures latex. (org-catch-invisible-edits 'show-and-error) (org-special-ctrl-a/e t) (org-insert-heading-respect-content t) - (org-hide-emphasis-markers t) (org-pretty-entities t) (org-agenda-tags-column 0) (org-ellipsis "…") @@ -518,7 +533,28 @@ This is my org mode configuration, which also configures latex. (python . t) (nix . t) (scheme . t) - (latex . t)))) + (latex . t))) + :config + (add-to-list 'org-preview-latex-process-alist + '(xetex-imagemagick + :programs ("xelatex" "convert") + :description "pdf > png" + :message "you need to install the programs: xelatex and imagemagick." + :image-input-type "pdf" + :image-output-type "png" + :image-size-adjust (1.0 . 1.0) + :latex-compiler ("xelatex -interaction nonstopmode -output-directory %o %f") + :image-converter ("convert -density %D -trim -antialias %f -quality 100 %O"))) + + ;; Set this new process as your default + (setq org-preview-latex-default-process 'xetex-imagemagick) + (set-face-attribute 'org-document-title nil :height 1.5 :weight 'bold) + (set-face-attribute 'org-level-1 nil :height 1.4 :weight 'bold) + (set-face-attribute 'org-level-2 nil :height 1.3 :weight 'bold) + (set-face-attribute 'org-level-3 nil :height 1.2 :weight 'semibold) + (set-face-attribute 'org-level-4 nil :height 1.1 :weight 'normal) + (set-face-attribute 'org-level-5 nil :height 1.0 :weight 'normal) + (set-face-attribute 'org-level-6 nil :height 1.0 :weight 'normal)) (use-package org-tempo :after org) @@ -923,6 +959,11 @@ First, some small configurations and some evil-mode initilaization because I lik (matrix-org "matrix.org" "8448") (gimp-org "irc.gimp.org" "6697")) + (general-define-key + "C-=" #'text-scale-increase + "C--" #'text-scale-decrease + "C-0" #'text-scale-set) + (leader-key 'normal "o c" '(org-capture :wk "Capture") ;; Org Mode @@ -1820,6 +1861,13 @@ I use tabs because sometimes I like using the mouse (it's actually more efficien :config (pulsar-global-mode 1)) #+end_src +** VTerm +#+begin_src emacs-lisp +(use-package vterm + :custom + (vterm-kill-buffer-on-exit nil) + :hook (vterm-exit-functions . rp/vterm-close-window-on-exit)) +#+end_src * Unpinned ** Lean4 For some reason, lean4-mode is not in MELPA currently so I have to do this ugly thing: diff --git a/config/nix.org b/config/nix.org index dca7f38..195a28f 100644 --- a/config/nix.org +++ b/config/nix.org @@ -37,7 +37,7 @@ in "continuity" "spontaneity" "installer" - "rpi-zero" + # "rpi-zero" ]; } #+end_src @@ -255,7 +255,11 @@ and now for the main flake: name = "services-test-${hostname}"; nodes = { "${hostname}" = { ... }: { - _module.args = attrs // { isIntegrationTest = true; }; + _module.args = attrs // { + isIntegrationTest = true; + system = getSystem hostname; + monorepoSelf = null; + }; imports = mkHostModules hostname ++ [ "${nixpkgs}/nixos/modules/misc/nixpkgs/read-only.nix" { @@ -902,32 +906,35 @@ underlying interface and it breaks significantly less often. pulse.enable = lib.mkDefault config.monorepo.profiles.pipewire.enable; jack.enable = lib.mkDefault config.monorepo.profiles.pipewire.enable; wireplumber.enable = lib.mkDefault config.monorepo.profiles.pipewire.enable; + extraConfig = { - pipewire."92-low-latency" = { + pipewire."92-clock" = { "context.properties" = { "default.clock.rate" = 48000; - "default.clock.quantum" = 512; - "default.clock.min-quantum" = 512; - "default.clock.max-quantum" = 1024; + "default.clock.allowed-rates" = [ 48000 ]; + + "default.clock.quantum" = 2048; + "default.clock.min-quantum" = 1024; + "default.clock.max-quantum" = 4096; + + "default.clock.quantum-limit" = 8192; }; - pipewire-pulse."92-low-latency" = { - "context.properties" = [ - { - name = "libpipewire-module-protocol-pulse"; - args = { }; - } - ]; - "pulse.properties" = { - "pulse.min.req" = "32/48000"; - "pulse.default.req" = "32/48000"; - "pulse.max.req" = "32/48000"; - "pulse.min.quantum" = "32/48000"; - "pulse.max.quantum" = "32/48000"; - }; - "stream.properties" = { - "node.latency" = "32/48000"; - "resample.quality" = 1; - }; + }; + + pipewire-pulse."92-obs-very-stable" = { + "pulse.properties" = { + "pulse.min.req" = "1024/48000"; + "pulse.default.req" = "2048/48000"; + "pulse.max.req" = "4096/48000"; + + "pulse.min.quantum" = "512/48000"; + "pulse.max.quantum" = "4096/48000"; + }; + + "stream.properties" = { + "node.latency" = "2048/48000"; + "node.max-latency" = "4096/48000"; + "resample.quality" = 10; }; }; }; @@ -1598,7 +1605,7 @@ Use ollama for serving large language models to my other computers. { config, lib, ... }: { services.bitcoind."${config.monorepo.vars.userName}" = { - enable = lib.mkDefault config.monorepo.profiles.workstation.enable; + enable = lib.mkDefault false; prune = 10000; }; } @@ -2575,6 +2582,7 @@ in "gcm" "sha256" "sha384" + "uvcvideo" ]; kernelParams = [ @@ -2820,23 +2828,39 @@ in tctiEnvironment.enable = true; }; - auditd.enable = true; - audit.enable = true; chromiumSuidSandbox.enable = (! config.monorepo.profiles.ttyonly.enable); sudo.enable = true; }; xdg.portal = { enable = (! config.monorepo.profiles.ttyonly.enable); - wlr.enable = (! config.monorepo.profiles.ttyonly.enable); + wlr = { + enable = (! config.monorepo.profiles.ttyonly.enable); + settings = { + screencast = { + chooser_type = "none"; + output_name = "DP-1"; + }; + }; + }; extraPortals = with pkgs; if (! config.monorepo.profiles.ttyonly.enable) then [ xdg-desktop-portal-gtk xdg-desktop-portal - xdg-desktop-portal-hyprland + xdg-desktop-portal-wlr ] else []; config.common.default = "*"; }; + systemd.user.services.xdg-desktop-portal-wlr = { + serviceConfig = { + Restart = lib.mkForce "on-failure"; + }; + environment = { + XDG_CURRENT_DESKTOP = "qtile"; + XDG_SESSION_TYPE = "wayland"; + }; + }; + environment.etc."gitconfig".text = '' [init] defaultBranch = main @@ -3292,7 +3316,7 @@ in supercollider inkscape kdePackages.kdenlive - kicad + # kicad ]) else []); monorepo.profiles = { @@ -3371,6 +3395,19 @@ These are some secrets that I use regularly for my programs in home. }; } #+end_src +*** OBS +#+begin_src nix :tangle ../nix/modules/home/obs.nix +{ pkgs, config, ... }: +{ + programs.obs-studio = { + enable = config.monorepo.profiles.workstation.enable; + plugins = with pkgs.obs-studio-plugins; [ + wlrobs + ]; + }; +} +#+end_src + *** Firefox I conditionally enable metamask based on the cryptocurrency option. Everything else here should be straightforward. @@ -3378,7 +3415,8 @@ be straightforward. { lib, config, pkgs, ... }: { programs.librewolf = { - enable = lib.mkDefault config.monorepo.profiles.graphics.enable; + # enable = lib.mkDefault config.monorepo.profiles.graphics.enable; + enable = false; package = pkgs.librewolf; profiles = { default = { @@ -3401,6 +3439,10 @@ be straightforward. }; } #+end_src + +#+RESULTS: +: <LAMBDA> + *** QuteBrowser #+begin_src nix :tangle ../nix/modules/home/qutebrowser.nix { lib, config, catppuccin-qutebrowser, ... }: @@ -3432,9 +3474,12 @@ be straightforward. # Hints fonts.hints = "bold 12pt Lora"; + + # Rendering + qt.force_software_rendering = "chromium"; }; extraConfig = (builtins.readFile "${catppuccin-qutebrowser}/setup.py") + -'' + '' config.load_autoconfig() setup(c, "mocha", True) ''; @@ -3641,10 +3686,11 @@ the timezone. My iamb profile. Note that iamb does not support calling (obviously, as it is a terminal app), but the nice thing about it is that I can set it up declaratively, so in case element-desktop stops working because of lack of declarative setup, I can still use this. #+begin_src nix :tangle ../nix/modules/home/iamb.nix -{ super, lib, config, ... }: +{ super, ... }: { programs.iamb = { - enable = lib.mkDefault config.monorepo.profiles.graphics.enable; + # enable = lib.mkDefault config.monorepo.profiles.graphics.enable; + enable = false; settings = { default_profile = "personal"; profiles.personal = { @@ -4089,6 +4135,12 @@ for these configurations. username = super.monorepo.vars.userName; homeDirectory = "/home/${super.monorepo.vars.userName}"; stateVersion = "24.11"; + sessionVariables = { + QTWEBENGINE_FORCE_USE_GBM = 0; + NIXOS_OZONE_WL = 1; + XDG_SESSION_TYPE = "wayland"; + XDG_CURRENT_DESKTOP = "qtile"; + }; packages = with pkgs; (if config.monorepo.profiles.graphics.enable then [ # wikipedia @@ -4109,11 +4161,11 @@ for these configurations. graphviz jq # Apps - octaveFull - grim swww vim element-desktop signal-desktop signal-cli thunderbird jami imv slurp + # octaveFull + grim swww vim element-desktop signal-desktop signal-cli thunderbird jami imv slurp wl-clipboard # Sound/media - pavucontrol alsa-utils imagemagick ffmpeg helvum pulseaudio + pavucontrol alsa-utils imagemagick ffmpeg pulseaudio # Net curl rsync gitFull ungoogled-chromium devd diff --git a/journal/20260331.org b/journal/20260331.org new file mode 100644 index 0000000..02a87a8 --- /dev/null +++ b/journal/20260331.org @@ -0,0 +1,9 @@ +#+TITLE: Daily Journal +#+STARTUP: showeverything +#+DESCRIPTION: My daily journal entry +#+AUTHOR: Preston Pan +#+date: +#+options: broken-links:t +* Tuesday, 31 March 2026 +** 21:56 +I'm calling my friend, and many people are gathered in my other friend's room. diff --git a/journal/20260402.org b/journal/20260402.org new file mode 100644 index 0000000..cf81c31 --- /dev/null +++ b/journal/20260402.org @@ -0,0 +1,9 @@ +#+TITLE: Daily Journal +#+STARTUP: showeverything +#+DESCRIPTION: My daily journal entry +#+AUTHOR: Preston Pan +#+date: +#+options: broken-links:t +* Thursday, 02 April 2026 +** 22:09 +Went out with a friend today and talked to some people in a listening party (Good Kid). diff --git a/journal/20260404.org b/journal/20260404.org new file mode 100644 index 0000000..f1033f3 --- /dev/null +++ b/journal/20260404.org @@ -0,0 +1,9 @@ +#+TITLE: Daily Journal +#+STARTUP: showeverything +#+DESCRIPTION: My daily journal entry +#+AUTHOR: Preston Pan +#+date: +#+options: broken-links:t +* Saturday, 04 April 2026 +** 23:21 +I'm almost finished this one complex analysis problem that has stumped me for a bit, which is very good. diff --git a/journal/20260411.org b/journal/20260411.org new file mode 100644 index 0000000..a38ca3f --- /dev/null +++ b/journal/20260411.org @@ -0,0 +1,9 @@ +#+TITLE: Daily Journal +#+STARTUP: showeverything +#+DESCRIPTION: My daily journal entry +#+AUTHOR: Preston Pan +#+date: +#+options: broken-links:t +* Saturday, 11 April 2026 +** 13:21 +I'm currently updating my monorepo again, and I'm doing more complex analysis. I think I get what tricks are important in it now. diff --git a/mindmap/Cauchy's Theorem.org b/mindmap/Cauchy's Theorem.org new file mode 100644 index 0000000..38f6c06 --- /dev/null +++ b/mindmap/Cauchy's Theorem.org @@ -0,0 +1,27 @@ +:PROPERTIES: +:ID: b243a8c0-ca7c-40e6-95b4-0f725a1a361f +:END: +#+title: Cauchy's Theorem +#+author: Preston Pan +#+description: Spinning around the complex plane. +#+options: broken-links:t + +* Introduction +Cauchy's theorem is the analogue of Green's Theorem for complex variables. It is a part of many equivalent statements made about analytic +functions. For example: +- exact differentials are closed. +- The harmonic conjugates of analytic functions satisfy the Cauchy-Riemann equations. +- Closed differentials describe [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative force]] fields. +- Harmonic functions satisfy Laplace's Equation. +- Under contour integration, the closed differentials are exactly those differentials which also satisfy the Cauchy-Riemann equations. +- A function is analytic iff it satisfies the Cauchy-Riemann equations. +- Analytic functions are conformal mappings except at their zeros. +and many more, are statements about the same set of objects, posed in different ways. +* Theorem +#+begin_theorem +If $D$ is a bounded domain with piecewise smooth boundary and $f$ is an analytic function which extends smoothly to $D \cup \partial D$, then $\oint_{D}f(z)dz = 0$. +#+end_theorem + +#+begin_proof +The closed differentials in the complex plane under contour integration are exactly those which satisfy the Cauchy-Riemann equations. +#+end_proof diff --git a/mindmap/Compactness.org b/mindmap/Compactness.org new file mode 100644 index 0000000..e175c43 --- /dev/null +++ b/mindmap/Compactness.org @@ -0,0 +1,14 @@ +:PROPERTIES: +:ID: 72deb4cd-46f7-4ef2-9c66-6943e47a9e83 +:ROAM_ALIASES: "open cover" compact compactness +:END: +#+title: Compactness +#+author: Preston Pan +#+description: Basic analysis and topology. +#+options: broken-links:t +* Introduction +A compact [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]] is a topological space such that every open cover has a finite subcover. That is, if $\mathbb{U}$ is a collection of open +sets $U$ that cover $X$, then there exists a subset $V$ of $\mathbb{U}$ such that $V$ is finite and covers $X$. + +An equivalent definition is that of in terms of [[id:d6dd23da-78be-420f-9103-4a81745aa272][nets]]; a set is compact if and only if all [[id:d6dd23da-78be-420f-9103-4a81745aa272][universal nets]] converge. We will prove this in this article, +as well as several basic properties and definitions related to compactness. diff --git a/mindmap/Proper Mapping.org b/mindmap/Proper Mapping.org new file mode 100644 index 0000000..633ce5e --- /dev/null +++ b/mindmap/Proper Mapping.org @@ -0,0 +1,14 @@ +:PROPERTIES: +:ID: 86bab66a-6f30-4330-966f-3ac319344602 +:ROAM_ALIASES: "proper map" +:END: +#+title: Proper Mapping +#+author: Preston Pan +#+description: It's proper and it's a map. +#+options: broken-links:t +* Introduction +Here is the definition: +#+begin_definition +If $f$ is a [[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][mapping]] on a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topological space]] $X$, then $f$ is proper if for all [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] sets $K \subset X$, $f^{-1}(K)$ is compact. +#+end_definition +We care about this definition because for some reason it is useful sometimes. diff --git a/mindmap/Separation Axioms.org b/mindmap/Separation Axioms.org new file mode 100644 index 0000000..a924c32 --- /dev/null +++ b/mindmap/Separation Axioms.org @@ -0,0 +1,51 @@ +:PROPERTIES: +:ID: 92d8e7ce-1008-43fb-ba7e-a36698a29fed +:ROAM_ALIASES: "separation axioms" +:END: +#+title: Separation Axioms +#+author: Preston Pan +#+description: Top 10 separated spaces you NEED to know! +#+options: broken-links:t + +* Definitions +The separation axioms of a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topological space]] are definitions that are useful for discussing how different points and sets in a topology are separated +from each other. In ascending order of strength we list them here. +** Kolmogorov Space (T0) +:PROPERTIES: +:ID: eab0e9c0-3fae-4870-840b-2a88a2deb215 +:ROAM_ALIASES: "T0 space" "Kolmogorov Space" +:END: +A space where for all $x, y \in X$, there exists $U$ such that $x \in U$ yet $y \not \in U$ OR vise versa. Also called /distinguishable/ or T0. You might think +these are useless, but notably, /any/ topological space can be converted into a T0 space by factoring out indistinguishable points. +** T1 Space +:PROPERTIES: +:ID: 954e6ba0-d655-412e-accd-d78c965b7f97 +:END: +A space where for all $x, y \in X$ there exists $U, V$ such that $x \in U, y \in V$ yet $x \not \in V$, $y \not \in U$. These spaces are interesting because +singletons are closed. For example take any singleton $\lbrace x \rbrace$ and consider the open set $\cup_{y \not = x} U_{y}$ where each [[id:e4ac2e89-1975-40de-9d6a-98281a3ca83e][open neighborhood]] of $y$ $U_{y}$ does not contain +$x$. The complement of this set is closed, and is precisely the singleton. +** Hausdorff (T2) +:PROPERTIES: +:ID: deb370a5-41a3-4ae5-b83f-4ba65ca71e29 +:ROAM_ALIASES: "Hausdorff Space" +:END: +A space where for all $x, y \in X$, there exists $U$, $V$ such that $x \in U$, $y \in V$, yet $U \cap V = \emptyset$. Notably [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][limits]] on [[id:d6dd23da-78be-420f-9103-4a81745aa272][nets]] converge uniquely when +they converge in these Hausdorff spaces. +** Regular (T3) +:PROPERTIES: +:ID: 01fa23a6-9a0d-4a28-ac82-2bcbb4e26a5c +:END: +A space where for all $x \in X$ and closed sets $F \subset X$ such that $x \not \in X$, there are open sets separating $F$ and $x$ in the same sense that they +separate points in the Hausdorff spaces. Yet, it is possible for regular spaces under this definition to be not strictly stronger than Hausdorff +spaces. For instance, not all singletons are closed in any topology. Therefore in order to restore the total ordering in terms of separation axiom +strength, most people also define regular spaces to have to be [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Spaces]] as well. From here on out we will in general assume that these spaces are Hausdorff. +** Tychonoff Space (T3.5) +:PROPERTIES: +:ID: 0ac540c2-9707-415a-b628-f2f01d73788c +:ROAM_ALIASES: "completely regular" +:END: +A space where for all $x \in X$ and closed sets $F \subset X$ such that $x \not \in X$, there is a [[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][continuous function]] $f: X \rightarrow [0, 1]$ that separates $x$ and $F$ +such that $f(x) = 0$ and $f(F) \equiv 1$ (every point in $F$ maps to $1$). this property is interesting because of its theoretical importance in the +[[id:14bebb09-2e38-4b55-adc0-97ba571331af][Stone-Cech Compactification]]. Also called /completely regular./ +** Normal (T4) +A space where for all closed $F, G \subset X$, there exists open sets $U, V$ separating them. This property is useful for applying Urysohn's Lemma. diff --git a/mindmap/Tychonoff's Theorem.org b/mindmap/Tychonoff's Theorem.org new file mode 100644 index 0000000..af12d8b --- /dev/null +++ b/mindmap/Tychonoff's Theorem.org @@ -0,0 +1,24 @@ +:PROPERTIES: +:ID: 80901a90-7ffd-4b86-9619-c8a71f4a2a72 +:END: +#+title: Tychonoff's Theorem +#+author: Preston Pan +#+description: +#+options: broken-links:t +* Introduction +Tychonoff's theorem is of great importance when dealing with the study of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]], and has far reaching results in the construction of the +[[id:14bebb09-2e38-4b55-adc0-97ba571331af][Stone-Cech Compactification]], notable for its universal property. +#+begin_theorem +The product of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topological spaces]] is compact. +#+end_theorem + +#+begin_proof +Let $X = \prod_{\alpha \in A} X_{\alpha}$ be a Tychonoff space. We will use the [[id:d6dd23da-78be-420f-9103-4a81745aa272][universal nets]] definition of compactness to prove $X$ is compact. +Also, we use the fact that the [[id:d6dd23da-78be-420f-9103-4a81745aa272][net]] $\lbrace x_{\beta} \rbrace$ converges in $X$ iff each of its projections $\pi_{\alpha} (x_{\beta})$ converges. + +If $f$ is a continuous mapping and $\lbrace x_{\beta} \rbrace$ is a universal net, then $f(x_{\beta})$ is universal. Therefore, because $\pi_{\alpha}$ is continuous for all +$\alpha$, and beacuse $X$ is compact for all $\alpha$, we conclude that for all universal nets $\lbrace x_{\beta} \rbrace$, the projections $\pi_{\alpha}(x_{\beta})$ converge for +all $\alpha$, and thus $\lbrace x_{\beta} \rbrace$ converges. +#+end_proof +Note that we are proving that the product of an /arbitrary family/ of compact spaces is compact, which makes the task seem a lot less difficult than it +is. Still, universal nets make the proof nice and easy. A special case is where all $X_{\alpha} = [0, 1]$. diff --git a/mindmap/directed set.org b/mindmap/directed set.org new file mode 100644 index 0000000..3b3d6a1 --- /dev/null +++ b/mindmap/directed set.org @@ -0,0 +1,59 @@ +:PROPERTIES: +:ID: 2517cbfe-bd7b-474f-993d-d4ee3c65a069 +:END: +#+title: Directed Set +#+author: Preston Pan +#+description: Central in order theory. +#+options: broken-links:t + +* Definition +A directed set $D$ is a set with some preorder defined on it: +\begin{align} + \forall \alpha, \beta \in D, \exists \gamma, \alpha \le \gamma, \beta \le \gamma +\end{align} +where $\ge$ obeys the usual rules for preorders (by convention, when we say $\alpha \le \gamma$ we are saying $\gamma \ge \alpha$). Though we will just use partial order +notation because the theory is equivalent if you just factor out by some equivalence relation. +* Nets +:PROPERTIES: +:ID: d6dd23da-78be-420f-9103-4a81745aa272 +:ROAM_ALIASES: net "universal net" +:END: +This notion is central to the study of compactness in the way that [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][sequences]] are. A net is a [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] $f: D \rightarrow X$ which maps directed set elements into +members of a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]]. There is one main theorem regarding nets that are of central importance, which is that /every net has a universal +subnet/. This mirrors the [[id:1e484e9f-cfd5-48f7-a920-c242f732b452][Bolzano-Weierstrass Theorem]] in sequences, and has deep implications for [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]]. We will give an explanation of +universality as well as some definitions to aide the explanation. +** Common Definitions +These are some common definitions for nets which are used in [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] to define abstracted notions of convergence and [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]]. +*** Frequently +:PROPERTIES: +:ID: 222f5770-d618-4620-8bc0-5f7c1171f417 +:ROAM_ALIASES: frequently +:END: +#+begin_definition +A net $\lbrace x_{\alpha} \rbrace$ is /frequently/ in some set $A$ if for all $\alpha \in D$, there exists $\beta \in D$ such that $\beta \ge \alpha, x_{\beta} \in A$. +#+end_definition +*** Eventually +:PROPERTIES: +:ID: 18a8e850-963d-4cfc-810a-6568ec33b6af +:ROAM_ALIASES: eventually +:END: +#+begin_definition +A net $\lbrace x_{\alpha} \rbrace$ is /eventually/ in some set $A$ if there exists $\alpha \in D$ such that for all $\beta \ge \alpha$, $x_{\beta}\in A$. +#+end_definition +Often this definition is used as a shorthand in order to +** Universal Nets +Universal nets are defined as nets that are /either/ [[id:18a8e850-963d-4cfc-810a-6568ec33b6af][eventually]] in $A$ or eventually in $A^{c}$ for all $A$ in a topological space $X$. Clearly, they are +of great importance to the study of both order theory and [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]]. The main theorem is this: +#+begin_theorem +every net has a universal subnet. +#+end_theorem + +#+begin_proof +Use Zorn's lemma or the Axiom of choice. +#+end_proof +and can be used to prove Tychonoff's theorem, a main result in the study of [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Spaces]]. +* Pitfalls +Note these couple facts: +- subnets of sequences are not always sequences! Subnets can branch, repeat, and use entirely different directed sets. The only requirement is that + subnets preserve order. +- nets don't converge uniquely in general; only when the space is a [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Space]] do nets converge uniquely when they /do/ converge. diff --git a/mindmap/duality.org b/mindmap/duality.org index f3267bf..96cd0a0 100644 --- a/mindmap/duality.org +++ b/mindmap/duality.org @@ -1,5 +1,6 @@ :PROPERTIES: :ID: 1b1a8cff-1d20-4689-8466-ea88411007d7 +:ROAM_ALIASES: dual :END: #+title: duality #+author: Preston Pan diff --git a/mindmap/function.org b/mindmap/function.org index 5ed1efb..4343848 100644 --- a/mindmap/function.org +++ b/mindmap/function.org @@ -4,7 +4,6 @@ #+title: function #+author: Preston Pan #+description: Not the best explanation, but it functions. - #+options: broken-links:t * Definition @@ -17,6 +16,18 @@ S = \{(x, y): x^{2} = y, x, y \in \mathbb{R} \} \end{align*} Which is an example of a parabolic function. \(x\) and \(y\) can both conceptually be any object, but usually they are mathematical objects. Some examples of such objects include tensors and scalars. +** Surjectivity +:PROPERTIES: +:ID: de33062a-35cf-4eae-a6bb-38b76dd4faf3 +:ROAM_ALIASES: onto surjective +:END: +A function is /surjective/, or /onto/, if every element in the codomain is mapped onto by an element in the domain. +** Injectivity +:PROPERTIES: +:ID: c34ad97b-b536-40f2-91d1-cb2ce788628a +:ROAM_ALIASES: injective +:END: +A function is /injective/, or /one-to-one/, if every element in the domain maps to a unique element in the codomain. * ordered pair :PROPERTIES: :ID: 1b1b522e-d4de-4832-9ca4-c6d1cfee27e6 @@ -30,4 +41,4 @@ Where the element that is not explicitly a set gives us the definition of the fi * Function Group Let \((S, \circ)\) define a [[id:ba7b95b0-0ce6-4b33-9a79-5e5fddaea710][group]] where \(S\) is the set of all functions, and \(\circ\) is the composition binary operator. Then \(f(x) = x\) is the identity element, and an inverse of a function is defined -as \( (f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x \). +as \( (f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x \). This only works if all the functions in your group have inverses, obviously. diff --git a/mindmap/limit.org b/mindmap/limit.org index be543ad..718e43a 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -1,18 +1,19 @@ :PROPERTIES: :ID: 122fd244-ffeb-47d0-89ce-bf9bc6f01b70 +:ROAM_ALIASES: Sequence sequence :END: #+title: limit +#+date: 2026-04-01 #+author: Preston Pan #+description: Pushing math to its limit #+LATEX_HEADER: \usepackage{tikz-cd} - #+options: broken-links:t - * Introduction A limit in mathematics is a tool used to describe the intuitive process of a value or a set of values tending towards another. First, we will define -limits as they pertain to sequences, and then we will define them on [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][functions]]. -For a sequence $\{s_{n}\}$: +limits as they pertain to sequences, and then we will define them on [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][functions]]. A sequence is defined as a function $s: \mathbb{N} \rightarrow X$ where $X$ is +any set, but here we will be talking about $X$ either as a [[id:6f24f731-60e5-4904-88d7-c63869505981][metric space]] or as $\mathbb{R}^{n}$, generally, based on the context. +For a sequence $\{s_{n}\}$: \begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N , n > N \implies | s_{n} - s | < \epsilon @@ -21,72 +22,35 @@ For a sequence $\{s_{n}\}$: What this means is that at some point in the sequence, for some choice of epsilon, no matter how small it is, there has to be an index where every term after that index is closer to $s$ than epsilon. If some single number $s$ and sequence $\{s_{n}\}$ fulfills this criteria, then it is said that the limit -of the sequence is $s$. Generally speaking, we use the set $\mathbb{R} \cup \{ -\infty, +\infty \}$, where there is a natural +of the sequence is $s$. Generally speaking, we use the set $\mathbb{R} \cup \{ -\infty, +\infty \}$, where there is an ordering: \begin{align*} \forall a \in \mathbb{R}, - \infty < a < +\infty \end{align*} -defined. Note that we can define equivalence relations on these symbols, but algebra reamins undefined. -** Unbounded Sequences -Unbounded sequences can still limit to $+\infty$ or $-\infty$, although the limit does not exist -for many unbounded sequences. If a sequence is one of: -1. unbounded above -2. unbounded below -but not both, it is possible that such sequences limit to $\infty$. -** Limits on Monotone Sequences -An increasing sequence is a sequence $\{s_{n}\}$ defined such that: -\begin{align*} -\forall n \in \mathbb{N}, \forall m \in \mathbb{N}, n \ge m \implies s_{n} \ge s_{m}. -\end{align*} +defined. Note that ordering can be defined on these symbols but the algebra remains undefined. +An equivalent and perhaps more intuitive definition which is equivalent defines a sequence in terms of the [[id:e4ac2e89-1975-40de-9d6a-98281a3ca83e][open neighbourhoods]] of a point. In +particular, a sequence $\lbrace s_{n} \rbrace$ converges to $s$ if and only if it is eventually in every open neighbourhood of $s$. -#+begin_theorem -The limit of monotone sequences always exists. -#+end_theorem - -#+begin_proof -We know: -\begin{align*} -\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ -\end{align*} -which is equivalent to: -\begin{align*} -\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies s - \epsilon < s_{n} < s + \epsilon -\end{align*} -and our sequence $\{s_{n}\}$ is monotone. If $\{s_{n}\}$ is increasing, we have: -\begin{align*} -s_{n + 1} \ge s_{n} -\end{align*} -for all n. Without loss of generality we shall assume $\{s_{n}\}$ is increasing. Then we take two cases: -1. $\{s_{n}\}$ is bounded. -2. $\{s_{n}\}$ is unbounded. -In the case $\{s_{n}\}$ is bounded: -\begin{align} -\label{} -\exists M, \forall n, s_{n} \le M \\ -s_{0} \le ... \le s_{n} \le s_{n + 1} \le s_{n + 2} \le ... \le M -\end{align} - -#+end_proof -** Limits as Objects -Limits can also be objects. This is most aptly demonstrated in more abstract fields of mathematics such as algebraic topology, -where the central "object of importance" (a common theme in math is one where you have an object of importance) is the net. -Specifically, the limits of universal nets have a deep relation to compactness, but here we will explore the most informative and essential -form of this idea and its algebraic properties. We will quickly go over the one-point compactification, and then introduce the stone-cech -compactification after. -** One Point Compactification +A sequence is just one kind of object that can have a limit. There are many other kinds of limits that operate on many different kinds of objects, yet +a prime example of a limit would be the limit on sequences, and we cannot examine the structure of limits without at least one example! Therefore, we +will sometimes link to external pages, but when the connection between different objects gets too intricate we will introduce the concepts inline. The +[[id:1e484e9f-cfd5-48f7-a920-c242f732b452][Bolzano-Weierstrass Theorem]] in particular demonstrates the concept of limits nicely. To prove this theorem with a more general method, we will first +introduce one-point compactification, and then we will introduce theorems relating specifically to [[id:6f24f731-60e5-4904-88d7-c63869505981][metric spaces]]. +* One Point Compactification :PROPERTIES: :ID: 339b32e7-ad89-40d7-8b11-5b293bd1056f +:ROAM_ALIASES: sequence :END: The one-point compactification is the simplest possible compactification of a topological space as you are adding only one point, and it does have a rather simple definition, although it is really only interesting in locally compact hausdorff spaces. -Let $X$ be a locally compact Hausdorff space, then its one-point compactification is $X \cup \lbrace \infty \rbrace$, where the topology defined on +Let $X$ be a [[id:e0c63828-18a6-48b1-a3ad-3126a9b78102][locally compact Hausdorff]] space, then its one-point compactification is $X \cup \lbrace \infty \rbrace$, where the [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] defined on this is as follows: 1. if $U$ is open in $X$, $U$ is open in $X \cup \lbrace \infty \rbrace$. 2. if $F \subset X$ is a compact subset and $\infty \in F^{c}$, then $F^{c}$ is open. -The topology generated by these open sets it the topology associated with the one-point compactification of $X$. If $X$ is locally compact hausdorff, -then in fact this topology is compact hausdorff, which is why it is the notable case. We shall see this in a proof. +The [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] generated by these open sets it the topology associated with the one-point compactification of $X$. If $X$ is locally compact hausdorff, +then in fact this topology is a [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:deb370a5-41a3-4ae5-b83f-4ba65ca71e29][Hausdorff Space]], which is why it is the notable case. We shall see this in a proof. #+begin_theorem If $X$ is a locally compact Hausdorff space and $X^{\plus} = X \cup \lbrace \infty \rbrace$ is the one-point compactification of $X$, then $X^{\plus}$ is a compact Hausdorff space. @@ -95,9 +59,9 @@ compact Hausdorff space. #+begin_proof In order to prove this, we must first prove it is compact, then we must prove it is Hausdorff. For the first we will use proof by contradiction. Let $\lbrace x_{\alpha}\rbrace$ be a universal net in -$X^{\plus}$, then suppose $\lbrace x_{\alpha}\rbrace$ does not converge in $X^{\plus}$. Then $\lbrace x_{\alpha} \rbrace$ also doesn't converge to $\infty$, and let $U_{\infty}$ be an open -neighborhood of $\infty$ which $\lbrace x_{\alpha} \rbrace$ is not eventually in. Then the complement $U_{\infty}^{c}$ must be compact (the only way to define a -neighborhood of $\infty$ is in terms of the complements of compact sets). But if $\lbrace x_{\alpha} \rbrace$ is eventually in $U_{\infty}^{c}$ it is eventually in a compact +$X^{\plus}$, then suppose $\lbrace x_{\alpha}\rbrace$ does not converge in $X^{\plus}$. Then $\lbrace x_{\alpha} \rbrace$ also doesn't converge to $\infty$, and let $U_{\infty}$ be an +[[id:e4ac2e89-1975-40de-9d6a-98281a3ca83e][open neighbourhood]] of $\infty$ which $\lbrace x_{\alpha} \rbrace$ is not eventually in. Then the complement $U_{\infty}^{c}$ must be compact (the only way to define an +open neighbourhood of $\infty$ is in terms of the complements of compact sets). But if $\lbrace x_{\alpha} \rbrace$ is eventually in $U_{\infty}^{c}$ it is eventually in a compact set and must converge. However $\lbrace x_{\alpha} \rbrace$ is universal and therefore must eventually be in either $U_{\infty}$ (impossible by construction) or $U_{\infty}^{c}$ (also impossible). Contradiction! @@ -109,18 +73,55 @@ $\overline{U}$ is compact, and this set exists due to locally compact property o Importantly, the one-point compactification can be thought of as a generalisation of the compactification of $\mathbb{R}^n$ via identification with $S^n$, and it can be thought of as undoing stereographic projection. It is also the smallest possible compactification as you are only adding one point. Note that it is possible for $X$ itself to be compact, and in that case $\infty$ is a disconnected component. -** Stone-Cech Compactification + +Note that it is useless to talk about the compactification without some connection to extensions of [[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][mappings]], specifically to the new point we're +adding, $\lbrace \infty \rbrace$. It turns out that this extension is /unique/ for a class of [[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][mappings]] called [[id:86bab66a-6f30-4330-966f-3ac319344602][proper maps]]. +** Bolzano-Weierstrass Theorem +:PROPERTIES: +:ID: 1e484e9f-cfd5-48f7-a920-c242f732b452 +:END: +We shall prove a general result that will automatically prove the Bolzano Weierstrass theorem, which is a bit more generalisable as an +intuition/concept than the Bolzano-Weierstrass theorem. +#+begin_theorem +if $\lbrace s_{n} \rbrace$ is a sequence in a [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]] [[id:6f24f731-60e5-4904-88d7-c63869505981][metric space]] , then it has a convergent subsequence. +#+end_theorem + +#+begin_proof +For all $m \in \mathbb{N}$, we can cover $X$ with open balls $B(x, \frac{1}{m})$ for all $x \in X$, starting from $m = 1$. Take +a finite subcover $\mathbb{U}_{0}$ , then $\lbrace s_{n} \rbrace_{0} = \lbrace s_{n} \rbrace$ is clearly [[id:222f5770-d618-4620-8bc0-5f7c1171f417][frequently]] in at least one of these open sets. For +all $m \in \mathbb{N}$ take a subsequence $\lbrace s_{n} \rbrace_{m}$ of $\lbrace s_{n} \rbrace_{m-1}$ such that $\lbrace s_{n}\rbrace$ is in +some $B(x, \frac{1}{m})$, by taking covers and finite subcovers $U_{m}$. Then define a sequence $y_{n}$ such that $y_{m} = \lbrace s_{m} \rbrace_{m}$, which is eventually +in every neighbourhood of some $x \in X$, and thus converges. +#+end_proof +and finally we get the Bolzano-Weierstrass theorem for $\mathbb{R}^{n}$ for free, as $\mathbb{R}^{n} \cup \lbrace \infty\rbrace$ is a compact metric space: +#+begin_corollary +Every sequence in $\mathbb{R}^{n}$ either has a convergent subsequence, or has a subsequence that escapes to $\infty$. +#+end_corollary +Also, for the two-point compactification, it yields this result as well if you're working in that space: +#+begin_corollary +Every sequence in $\mathbb{R}$ has a subsequence that either converges in $\mathbb{R}$ or converges to one of $-\infty$, or $\infty$. +#+end_corollary +Also note that the proof above demonstrates the concept of /diagonalisation/, which is central in themes of /completion/ or +compactification. Specifically, using diagonal arguments in order to construct or complete, or show the completeness of a space is a central theme in +this branch of mathematics. +* Limits as Objects +Limits can also be objects. This is most aptly demonstrated in more abstract fields of mathematics such as algebraic topology, +where the central "object of importance" (a common theme in math is one where you have an object of importance) is the net. +Specifically, the limits of [[id:d6dd23da-78be-420f-9103-4a81745aa272][universal nets]] have a deep relation to [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compactness]], but here we will explore the most informative and essential +form of this idea and its algebraic properties. We will quickly go over the one-point compactification, and then introduce the stone-cech +compactification after. +* Stone-Cech Compactification :PROPERTIES: :ID: 14bebb09-2e38-4b55-adc0-97ba571331af :END: -We can construct the Stone Cech Compcatification on a completely regular topological space $X$, which will require a specific construction +We can construct the Stone Cech Compcatification on a [[id:0ac540c2-9707-415a-b628-f2f01d73788c][completely regular]] topological space $X$, which will require a specific construction but will at least give us the Hausdorff property in the compactified space. To start, let $A$ be the set of all $f_{\alpha}: X \rightarrow [0, 1]_{\alpha}$ such that $f$ is -continuous (with $\alpha$ being an arbitrary but consistent index), and let us define a Tychonoff space $Y = \prod_{\alpha \in A}[0, 1]_{\alpha}$ and an embedding $\phi: X \rightarrow Y$ +[[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][continuous]] (with $\alpha$ being an arbitrary but consistent index), and let us define a [[id:0ac540c2-9707-415a-b628-f2f01d73788c][completely regular space]] $Y = \prod_{\alpha \in A}[0, 1]_{\alpha}$ and an embedding $\phi: X \rightarrow Y$ where the embedding $\phi$ is defined as $(\phi(x))_{\alpha }= f_{\alpha}(x)$. Then the idea is that the /closure/ of $\phi(X)$ in $Y$ is a compactification of $X$. In fact, this is sort of analogous to currying in the theory of computer science, or delayed or /lazy evaluation/, and as we shall see, it will share similar algebraic properties. -How do we know the space is compact? We know that $Y$ is compact because $[0, 1]$ is compact, and we apply Tychonoff's theorem. How do we know that +How do we know the space is [[id:72deb4cd-46f7-4ef2-9c66-6943e47a9e83][compact]]? We know that $Y$ is compact because $[0, 1]$ is compact, and we apply [[id:80901a90-7ffd-4b86-9619-c8a71f4a2a72][Tychonoff's Theorem]]. How do we know that $\overline{\phi(X)}$ is compact? It is closed and a subset of a compact set. However, what we have /not/ shown thus far is that $\phi(X)$ is truly an embedding. To see this, the completely regular property of $X$ saves the day; if we /didn't/ have this property, then it would be possible for some two points to /never/ be separated by any function, and then you'd lose the one-to-one property of $\phi$. Also, $\phi$ is clearly always continuous; we use the property that @@ -139,21 +140,25 @@ if $X$ is a completely regular space and $\phi: X \rightarrow \beta X$ is the ev #+end_theorem #+begin_proof -In this proof we will use the net definition of continuity. Suppose $\phi(x_{\alpha}) \rightarrow \phi(x)$, yet $x_{\alpha} \not \rightarrow x$. Then there exists some open neighborhood $U$ of +In this proof we will use the net definition of continuity. Suppose $\phi(x_{\alpha}) \rightarrow \phi(x)$, yet $x_{\alpha} \not \rightarrow x$. Then there exists some [[id:e4ac2e89-1975-40de-9d6a-98281a3ca83e][open neighbourhood]] $U$ of $x$ such that $x_{\alpha}$ is not eventually in $U$. Because of complete regularity, there exists a map $f$ separating $x$ from $U^{c}$. If $\phi(x_{\alpha}) \rightarrow \phi(x)$, then $\pi_{f} \circ \phi(x_{\alpha}) \rightarrow \pi_{f} \circ \phi(x)$, but clearly this is equivalent to $f(x_{\alpha}) \rightarrow f(x)$. Any subnet of a convergent net converges to the same value, so we create a subnet $\lbrace y_{\alpha}\rbrace$ of $\lbrace x_{\alpha} \rbrace$ such that $\lbrace y_{\alpha}\rbrace$ is eventually in $U^{c}$ (this is possible because $\lbrace x_{\alpha}\rbrace$ is frequently in $U^{c}$). Then $f(y_{\alpha}) \rightarrow 1$ ($f(U^{c}) \equiv 1$ by construction), yet $f(x) = 0$. But this is clearly absurd, because $\lbrace y_{\alpha} \rbrace$ converges uniquely, and the constant $1$ net cannot converge to $0$! Contradiction. #+end_proof -*** Algebra on Limits +** Algebra on Limits Often times it is useful to think of limits as /objects in themselves/ rather than an object that you apply to, say, a sequence. Often times algebras on -different /kinds/ of limits enables oneself to draw on connections between limits and many other fields of mathematics. For instance, the /closure/ of a +different /kinds/ of limits enables oneself to draw on connections between limits and many other fields of mathematics. For instance, the [[id:1954ee72-ffce-4586-ad8a-a46c39c8f77d][closure]] of a set is exactly the same set with all its limit points included, and both closures, and as we will see, limits, are /idempotent/, which is to say, applying them once is the same thing as applying them twice. Note that if $f: X \rightarrow Y$ where $Y$ is any topological space and $f$ is any continuous -function, then $\beta f(X) = f(\beta X)$, which one can represent with a commutative diagram, where $\beta f$ is the /unique extension/ of the mapping $f$. Actually, in a moment +function, then $\beta f(X) = f(\beta X)$, which one can represent with a commutative diagram, where $\beta f$ is the /unique extension/ of the [[id:fdcecb13-35e1-439c-ba13-5c63bd7342c3][mapping]] $f$. Actually, in a moment we will see that the functor commuting is equivalent to the /limit/ commuting on nets. - -*** The Universal Property -We say the following diagram commutes: - +* I'm Here For Sequences Dude +Oh, sorry. In that case we can apply our learnings above for the purpose of giving you some concrete examples👍. +** Limits on Reals/Complex Numbers +I am pretty sure I already did this one above, but basically in $\mathbb{R}^{n}$ a limit converges iff each projection converges, in the same way as for +product spaces in general. Complex numbers are a product space. +** Limits on Functions +You can limit functions pointwise. What that means is for each $x$, you just do the limit thing. Also more importantly there is /uniform convergence/, +but I mean, that's a measure theory thing, and that's lame. diff --git a/mindmap/locally compact Hausdorff.org b/mindmap/locally compact Hausdorff.org new file mode 100644 index 0000000..5b9488a --- /dev/null +++ b/mindmap/locally compact Hausdorff.org @@ -0,0 +1,11 @@ +:PROPERTIES: +:ID: e0c63828-18a6-48b1-a3ad-3126a9b78102 +:END: +#+title: Locally Compact Hausdorff +#+author: Preston Pan +#+description: the LCH king will rise again +#+options: broken-links:t + +* Introduction +A locally compact Hausdorff space is a Hausdorff space in which every point has a [[id:e4ac2e89-1975-40de-9d6a-98281a3ca83e][compact neighbourhood]]. In particular, the [[id:339b32e7-ad89-40d7-8b11-5b293bd1056f][One Point Compactification]] +of a locally compact Hausdorff space is compact Hausdorff, so they are interesting to study for that reason. diff --git a/mindmap/mapping.org b/mindmap/mapping.org new file mode 100644 index 0000000..dbc1995 --- /dev/null +++ b/mindmap/mapping.org @@ -0,0 +1,11 @@ +:PROPERTIES: +:ID: fdcecb13-35e1-439c-ba13-5c63bd7342c3 +:ROAM_ALIASES: continuous mapping "continuous function" +:END: +#+title: Mapping +#+author: Preston Pan +#+description: mapping the thought landscape out. +#+options: broken-links:t +* Introduction +A /mapping/, or /continuous function/ in a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topological space]] is defined as a [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] $f$ such that the preimage of an [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][open set]] is open, or $f^{-1}(U)$ +is open for all [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][open sets]] $U$. diff --git a/mindmap/metric space.org b/mindmap/metric space.org index 2609691..338686c 100644 --- a/mindmap/metric space.org +++ b/mindmap/metric space.org @@ -2,14 +2,13 @@ :ID: 6f24f731-60e5-4904-88d7-c63869505981 :ROAM_ALIASES: metric :END: -#+title: metric space +#+title: Metric Space #+author: Preston Pan #+description: The basis of modern analysis. - #+options: broken-links:t * Introduction -A metric space $(G, d)$ is a set with a metric $d(x,y): G \times G \rightarrow \mathbb{R}$ defined on members of the set. +A metric space $(X, d)$ is a [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][Topological Space]] with a metric $d(x,y): X \times X \rightarrow \mathbb{R}$ defined on members of the set. This metric is a generalization of distance, with the following properties: \begin{align} \label{} @@ -18,4 +17,6 @@ x \ne y \implies d(x, y) > 0 \\ d(x, y) = d(y, x) \\ d(x, z) \le d(x, y) + d(x, z) \end{align} -where property $(4)$ is the triangle inequality. +where property $(4)$ is the triangle inequality. Also, the metric generates the [[id:b0784577-9691-4c8e-a8e4-974a7c9c4949][topology]] on the open sets; a basis can be chosen by including every +open ball, which is defined as $B(x, r) = \lbrace y: d(x, y) < r\rbrace$. A neighbourhood basis can be chosen by including every open rational ball +that is a neighbourhood of $x$, and in fact this neighbourhood basis is countable, so metric spaces are first countable. diff --git a/mindmap/neighbourhood.org b/mindmap/neighbourhood.org new file mode 100644 index 0000000..f7014e5 --- /dev/null +++ b/mindmap/neighbourhood.org @@ -0,0 +1,15 @@ +:PROPERTIES: +:ID: e4ac2e89-1975-40de-9d6a-98281a3ca83e +:ROAM_ALIASES: "compact neighbourhood" "closed neighbourhood" "open neighbourhood" "open neighborhood" +:END: +#+title: Neighbourhood +#+author: Preston Pan +#+description: Locality in topological spaces. +#+options: broken-links:t + +* Introduction +A /neighbourhood/ $N$ of a point $x$ in a topological space $X$ is defined as a set in a topological space which contains an open set $U$ containing +the point $x$. + +Often times it is useful for defining [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][limit]] behavior on the topology. An /open neighbourhood/ is a neighbourhood which is also an open set, and a +compact and closed neighbourhood follows those same naming conventions, respectively. diff --git a/mindmap/stereographic projection.org b/mindmap/stereographic projection.org new file mode 100644 index 0000000..0e08d7e --- /dev/null +++ b/mindmap/stereographic projection.org @@ -0,0 +1,7 @@ +:PROPERTIES: +:ID: c7e123d2-b488-4aec-8b66-98782f1b5775 +:END: +#+title: stereographic projection +#+author: Preston Pan +#+description: as opposed to monographic projection. +#+options: broken-links:t diff --git a/mindmap/topological space.org b/mindmap/topological space.org new file mode 100644 index 0000000..01fe167 --- /dev/null +++ b/mindmap/topological space.org @@ -0,0 +1,27 @@ +:PROPERTIES: +:ID: b0784577-9691-4c8e-a8e4-974a7c9c4949 +:ROAM_ALIASES: "topological space" "open set" topology +:END: +#+title: Topological Space +#+author: Preston Pan +#+description: Algebraic? Geometric? Fantastic! +#+options: broken-links:t + +* Definition +A topological space is a set $X$, equipped with a topology. That is, it is equipped with a collection of subsets that are considered to be the /open +sets/ of that topology. These open sets must obey several rules: +1. $\cup_{\alpha \in A}U_{\alpha}$ is open, if all $U_{\alpha}$ are open. +2. $\cap_{n=0}^{N}U_{n}$ is open, if $N$ is finite and $U_{n}$ are open. +3. $\emptyset$ is open, and $X$ is open. +the [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][dual]] concept to open sets are closed sets, which are the complements of open sets. Note that closed sets can also be open sets, and vise versa; a +simple example is the space itself, in any topology; $X$ is open by definition, yet it is also closed because $\emptyset^{c} = X$. This is not just a trivial +example; these "clopen" sets are fairly common (this is in fact the terminology people use). +* More Basic Definitions +Here we introduce several more basic definitions so that we can talk about them in other articles. +** Closure +:PROPERTIES: +:ID: 1954ee72-ffce-4586-ad8a-a46c39c8f77d +:ROAM_ALIASES: interior closure +:END: +The /closure/ of a set $F$ in a topological space $X$ is denoted $\overline{F}$ and is defined as the smallest closed set which contains every open set +$U \subset F$. Likewise, the /interior/ of a set is defined as the largest open set which is inside $F$. diff --git a/nix b/nix -Subproject cff2184c501a59c8f2772f2fb20d0c8dd217cd9 +Subproject f33c1b08e066fafdaf6cf3ac10b66867451338c @@ -460,26 +460,33 @@ h1.title { vertical-align: middle } -.theorem, .proof { +.theorem, +.lemma, +.corollary, +.definition, +.proof { display: block; font-style: normal; background: color-mix(in srgb, var(--accent) 8%, transparent); border-left: 3px solid var(--accent); padding: 1rem 1.3rem; - margin-top: 2rem; - margin-left: 10px; - &::before { float: left; font-weight: bold; } + margin: 2rem 0 20px 10px; + &::before { + font-weight: bold; + margin-right: 0.4em; + } } -.theorem { margin-bottom: 20px; &::before { content: "Theorem.\00a0\00a0"; } } +.theorem::before { content: "Theorem."; } +.lemma::before { content: "Lemma."; } +.corollary::before { content: "Corollary."; } +.definition::before { content: "Definition."; } .proof { position: relative; margin-bottom: 30px; padding-right: 1.2em; - &::before { - content: "Proof.\00a0\00a0"; - } + &::before { content: "Proof.\00a0\00a0"; } &::after { content: ""; position: absolute; diff --git a/yasnippet/latex-mode/net b/yasnippet/latex-mode/net index 1a2f7b8..7ce63a4 100644 --- a/yasnippet/latex-mode/net +++ b/yasnippet/latex-mode/net @@ -2,4 +2,4 @@ # name: net # key: net # -- -$\lbrace $0_{\alpha} \rbrace$
\ No newline at end of file +\lbrace $0_{\alpha} \rbrace
\ No newline at end of file diff --git a/yasnippet/latex-mode/notin b/yasnippet/latex-mode/notin new file mode 100644 index 0000000..44a7715 --- /dev/null +++ b/yasnippet/latex-mode/notin @@ -0,0 +1,5 @@ +# -*- mode: snippet -*- +# name: notin +# key: notin +# -- +\not \in
\ No newline at end of file diff --git a/yasnippet/latex-mode/sequence b/yasnippet/latex-mode/sequence new file mode 100644 index 0000000..dd2bf2a --- /dev/null +++ b/yasnippet/latex-mode/sequence @@ -0,0 +1,5 @@ +# -*- mode: snippet -*- +# name: sequence +# key: sequence +# -- +\lbrace $0_{n} \rbrace
\ No newline at end of file diff --git a/yasnippet/latex-mode/tikzcd b/yasnippet/latex-mode/tikzcd new file mode 100644 index 0000000..09fd000 --- /dev/null +++ b/yasnippet/latex-mode/tikzcd @@ -0,0 +1,7 @@ +# -*- mode: snippet -*- +# name: tikzcd +# key: tikzcd +# -- +\begin{tikzcd} +$0 +\end{tikzcd}
\ No newline at end of file diff --git a/yasnippet/org-mode/corollary b/yasnippet/org-mode/corollary new file mode 100644 index 0000000..bb011e5 --- /dev/null +++ b/yasnippet/org-mode/corollary @@ -0,0 +1,7 @@ +# -*- mode: snippet -*- +# name: corollary +# key: corollary +# -- +#+begin_corollary +$0 +#+end_corollary
\ No newline at end of file diff --git a/yasnippet/org-mode/definition b/yasnippet/org-mode/definition new file mode 100644 index 0000000..8873335 --- /dev/null +++ b/yasnippet/org-mode/definition @@ -0,0 +1,7 @@ +# -*- mode: snippet -*- +# name: definition +# key: definition +# -- +#+begin_definition +$0 +#+end_definition
\ No newline at end of file diff --git a/yasnippet/org-mode/lemma b/yasnippet/org-mode/lemma new file mode 100644 index 0000000..119359d --- /dev/null +++ b/yasnippet/org-mode/lemma @@ -0,0 +1,7 @@ +# -*- mode: snippet -*- +# name: lemma +# key: lemma +# -- +#+begin_lemma +$0 +#+end_lemma
\ No newline at end of file diff --git a/yasnippet/org-mode/new-snippet b/yasnippet/org-mode/new-snippet new file mode 100644 index 0000000..b3f5d77 --- /dev/null +++ b/yasnippet/org-mode/new-snippet @@ -0,0 +1,5 @@ +# -*- mode: snippet -*- +# name: new-snippet +# key: new +# -- +hello world $0
\ No newline at end of file |