\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
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

Connectedness. Definition


Let \( X \) be a topological space. A [...] of \( X \) is a pair \( U, V \) of disjoint nonempty open subsets of \( X \) whose union is \( X \). The space \( X \) is said to be connected iff [...].

This definition is from Munkres. Leinster's definition feels a little less intuitively concrete.

Connectedness, formulation 2 (or now a lemma)

A space \( X \) is connected iff the only subsets of \( X \) that [...] are [...].

Again, from Munkres.