\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \newcommand{\E} {\mathrm{E}} \)
deepdream of a sidewalk
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)
Math and science::Topology

Closed subsets

Closed subsets, definition

Let \( X = (X, \mathcal{T}) \) be a topological space. A subset \( V \subseteq X \) is closed iff [...].

The below properties of closed sets can be derived.

Lemma. [hidden as it gives away the below cloze.]

Let \( X = (X, \mathcal{T}) \) be a topological space. Then

  1. Whenever \( V_1 \) and \( V_2 \) are closed subsets of \( X \), then [...] is also closed in X.
  2. Whenever \( (V_i)_{i\in I} \) is a family (finite or not) of closed subsets of \( X \), then [...] is closed in \( X \).
  3. [something] and [something] are closed subsets of \( X \).

This lemma follows from the application of de Morgan's laws to the definition of a topological space.