Topology Generated by Closed Sets

From ProofWiki
Jump to navigation Jump to search

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


Sources