### Table of Contents

## Compact topological spaces

Abbreviation: **KTop**

### Definition

A \emph{compact topological space} is a topological space $\mathbf{X}=\langle X,\Omega\rangle$ that is

\emph{compact}: every open cover has a finite subcover, i.e., $\forall\mathcal C\subseteq\Omega(\bigcup\mathcal C=X\Longrightarrow\exists n, \exists C_0,\ldots,C_{n-1}\in\mathcal C(C_0\cup\cdots\cup C_{n-1}=X))$

Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{X}$ and $\mathbf{Y}$ be compact topological spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:X\rightarrow Y$ that is a continuous: $\forall V\in\Omega_{\mathbf Y}(h^{-1}[Y]\in\Omega_{\mathbf X})$

### Definition

A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Subclasses

[[Compact Hausdorff topological spaces]]

### Superclasses

[[Topological spaces]]