\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \)

Connectedness Examples

• \( X = \mathbb{R} \setminus \{0\} \) is [something about connectedness].
• The space of rationals numbers \( \mathbb{Q} \), topologized as a subspace of \( \mathbb{R} \), is [something about connectedness].
• A discrete space with 2 or more points is [something about connectedness].
• A non-empty indiscrete space is [something about connectedness].
• An interval topologized as a subspace of \( \mathbb{R} \) is [something about connectedness].
• The space \( \mathbb{R}^d \) is [something about connectedness].
• The letter 'O' is [something about connectedness].
• The collection of connected sets in \( \mathbb{R} \) coincide preciesly with [...].