Proposition: Alternative Characterization of Topological Spaces

Let $X$ be a set. A set $\mathcal C$ of closed subsets of $X$ defines a topology on $X$, if $\mathcal C$ fulfills the following properties:

  1. The empty set \(\emptyset\) and the whole set \(X\) are closed (i.e. \({\emptyset\in\mathcal {C}}\) and \({X\in\mathcal {C}}\) ).
  2. The union of finitely many closed sets is also closed, i.e. if \(U_{1},\ldots ,U_{n}\in {\mathcal {C}}\), then also \(U_{1}\cup \ldots \cup U_{n}\in {\mathcal {C}}\).
  3. The intersection of arbitrarily many closed sets is again closed, i.e. with \(U_{i}\in {\mathcal {O}}\) for each \(i\in I\) (for an arbitrary index set \(I\)) we have also \(\bigcap _{i\in I}U_{i}\in {\mathcal {C}}.\)

In other words, the ordered pair $(X,\mathcal C)$ is a topological space.

Proofs: 1

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  1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition