Topology Generated by Closed Sets

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Theorem

Let $X$ be a set.

Let $\mathcal F$ be a set of subsets of $X$.

Suppose that

$\varnothing \in \mathcal F$ and
for every subsets $A$ and $B$ of $X$ if $A, B \in \mathcal F$, then $A \cup B \in \mathcal F$ and
for every subset $\mathcal G \subseteq \mathcal F$, $\bigcap \mathcal G \in \mathcal F$ and
$\tau = \left\{{\complement_X \left(A\right): A \in \mathcal F}\right\}$.

Then:

$T = \left( {X, \tau} \right)$ is topological space and
for every subset $A$ of $X$, $A$ is closed in $T$ if and only if $A \in \mathcal F$.

Proof


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