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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 $F$. |