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

# {{ 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 $X$.
{% enddefinition %}

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

{% theorem %}
Every closed subspace of a compact space is compact.
{% endtheorem %}

{% proof %}
{% endproof %}

{% theorem %}
Every compact subspace of a Hausdorff space is closed.
{% endtheorem %}

{% proof %}
{% endproof %}