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## Definition: Topological Space, Topology

A topological space is a set $$X$$, together with a subset $${\mathcal {T}}$$ of the power set of $$X$$, which has the following structural properties

1. We call all subsets $$U\subseteq X$$ belonging to $${\mathcal {T}}$$ open sets.
2. The empty set $$\emptyset$$ and the whole set $$X$$ are open (i.e. $${\emptyset\in\mathcal {T}}$$ and $${X\in\mathcal {T}}$$ ).
3. The intersection of finitely many open sets is also open, i.e. if $$U_{1},\ldots ,U_{n}\in {\mathcal {T}}$$, then allso $$U_{1}\cap \ldots \cap U_{n}\in {\mathcal {T}}$$.
4. The union of arbitrarily many open sets is again open, i.e. with $$U_{i}\in {\mathcal {T}}$$ for each $$i\in I$$ (for an arbitrary index set $$I$$) we have also $$\bigcup _{i\in I}U_{i}\in {\mathcal {T}}$$.

| | | | | Contributors: bookofproofs | References: [6907]

## 1.Definition: Hausdorff Space

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[6907] Brenner, Prof. Dr. rer. nat., Holger: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014