# compact pavings are closed subsets of a compact space

Recall that a paving $\mathcal{K}$ is compact^{} if every subcollection satisfying the finite intersection property has nonempty intersection^{}. In particular, a topological space^{} is compact (http://planetmath.org/Compact) if and only if its collection^{} of closed subsets forms a compact paving. Compact paved spaces can therefore be constructed by taking closed subsets of a compact topological space. In fact, all compact pavings arise in this way, as we now show.

Given any compact paving $\mathcal{K}$ the following result says that the collection ${\mathcal{K}}^{\prime}$ of all intersections of finite unions of sets in $\mathcal{K}$ is also compact.

###### Theorem 1.

Suppose that $\mathrm{(}K\mathrm{,}\mathrm{K}\mathrm{)}$ is a compact paved space. Let ${\mathrm{K}}^{\mathrm{\prime}}$ be the smallest collection of subsets of $X$ such that $\mathrm{K}\mathrm{\subseteq}{\mathrm{K}}^{\mathrm{\prime}}$ and which is closed under arbitrary intersections and finite unions. Then, ${\mathrm{K}}^{\mathrm{\prime}}$ is a compact paving.

In particular,

$$\mathcal{T}\equiv \{K\setminus C:C\in {\mathcal{K}}^{\prime}\}\cup \{\mathrm{\varnothing},K\}$$ |

is closed under arbitrary unions and finite intersections, and hence is a topology on $K$. The collection of closed sets defined with respect to this topology is ${\mathcal{K}}^{\prime}\cup \{\mathrm{\varnothing},K\}$ which, by Theorem 1, is a compact paving. So, the following is obtained.

###### Corollary.

A paving $\mathrm{(}K\mathrm{,}\mathrm{K}\mathrm{)}$ is compact if and only if there exists a topology on $K$ with respect to which $\mathrm{K}$ are closed sets and $K$ is compact.

Title | compact pavings are closed subsets of a compact space |
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Canonical name | CompactPavingsAreClosedSubsetsOfACompactSpace |

Date of creation | 2013-03-22 18:45:01 |

Last modified on | 2013-03-22 18:45:01 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 28A05 |