# Topology Generated by Closed Sets

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## Theorem

Let $X$ be a set.

Let $\FF$ be a set of subsets of $X$ such that:

- $\O \in \FF$
- $\forall A, B \in \FF: A \cup B \in \FF$
- $\forall \GG \subseteq \FF: \bigcap \GG \in \FF$

Let $\tau = \set {\relcomp X A: A \in \FF}$.

Then:

- $T = \struct {X, \tau}$ is topological space and
- for every subset $A$ of $X$, $A$ is closed in $T$ if and only if $A \in \FF$.

## Proof

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## Sources

- Mizar article TOPGEN_3:4