Limit Point Compactness

Definition

Consider a Topological Space $(X,\mathcal{T})$. The set $X$ is limit point compact if every infinite subset has a Limit Point.

Examples

Consider a Product Topology $\mathbb{Z}\times Y$ of the two topological spaces $(\mathbb{N},{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$ where $Y=\{a,b\}$ and ${\mathcal{T}}_Y=\{\varnothing ,\{a,b\}\}$. It is not Compact but limit point compact $\because$ Each element $\{n\}\times \{a\}$ is a Limit Point of $\{n\}\times \{b\}$ and vice versa by the construction of topology ${\mathcal{T}}_Y$

Facts

Consider a Topological Space $(X,\mathcal{T})$. If $X$ is compact then $X$ is Limit Point Compact

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Consider a metrizable space $(X,\mathcal{T})$. Then, $X$ is compact if and only if $X$ is Limit Point Compact if and only if $X$ is Sequentially Compact

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