# 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

Github: ### References

#### Bibliography

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