: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.