---
title: Basics
parent: Compactness
grand_parent: General Topology
nav_order: 1
published: false
# cspell:words
---

# {{ page.title }} of Compact Spaces

*Compact space* is short for compact topological space.

{: .definition }
> Suppose $X$ is a topological space.
> A *covering* of $X$ is a collection $\mathcal{A}$
> of subsets of $X$ such that
> $\bigcup \mathcal{A} = X$.
> A covering $\mathcal{A}$ of $X$ is called *open*
> if each member of the collection $\mathcal{A}$
> is open in $X$.
> A covering $\mathcal{A}$ is called *finite*
> the collection $\mathcal{A}$ is finite.
> A *subcovering* of a covering $\mathcal{A}$ of $X$
> is a subcollection $\mathcal{B}$ of $\mathcal{A}$
> such that $\mathcal{B}$ is a covering of $X$.

{: .definition }
> A topological space $X$ is called *compact*
> if every open covering of $X$
> has a finite subcovering.

{: .theorem }
> Every closed subspace of a compact space is compact.

{% proof %}
{% endproof %}

{: .theorem }
> Every compact subspace of a Hausdorff space is closed.

{% proof %}
{% endproof %}