## Compact topological spaces

Abbreviation: **KTop**

### Definition

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

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

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

$...$ is …: $axiom$

$...$ 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.

Classtype | second-order |
---|---|

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Subclasses

[[Compact Hausdorff topological spaces]]

### Superclasses

[[Topological spaces]]

### References

Trace: » compact_topological_spaces