Isolated Point

Definition

Consider a subspace topology $(A,{\mathcal{T}}^{\prime })$ of a Topological Space $(X,\mathcal{T})$, and a point $x\in A$. $x\in A$ is an isolated point if the singleton set $\{x\}$ is open in the Subspace Topology ($\{x\}\in {\mathcal{T}}^{\prime }$).

Or, equivalently, $x\in A$ is an isolated point if $\exists {\mathcal{N}}_x\text{s.t.}{\mathcal{N}}_x\cap (A\setminus \{x\})=\varnothing$ where ${\mathcal{N}}_x$ is the Neighborhood of $x$.

Facts

Consider a nonempty Compact Hausdorff Space $X$. If $X$ has no isolated point, then $X$ is uncountable.