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

Metric space. Open and closed sets

Open and closed subsets

Let \( X \) be a metric space.

Open in \( X \)
A subset \( U \) of \( X \) is open in \( X \) iff [...].
Closed in \( X \)
A subset \( V \) of \( X \) is closed in \( X \) iff [...].

Tom Leinster describes the openness of \( U \):

Thus, \( U \) is open if every point of \( U \) has some elbow room—it can move a little bit in each direction without leaving \( U \).

Personally, I like the phrase: every element of an open set has a neighbourhood.

ε-balls

Open ε-balls are open, and closed ε-balls are closed. Consider trying to prove this. They are open on account of the definition of openness, not by their own definition alone, despite their names being suggestive.