**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

- We call all subsets \(U\subseteq X\) belonging to \({\mathcal {T}}\)
**open sets**. - The empty set \(\emptyset\) and the whole set \(X\) are open (i.e. \({\emptyset\in\mathcal {T}}\) and \({X\in\mathcal {T}}\) ).
- The intersection of finitely many open sets is also open, i.e. if \(U_{1},\ldots ,U_{n}\in {\mathcal {T}}\), then allso \(U_{1}\cap \ldots \cap U_{n}\in {\mathcal {T}}\).
- 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 OsnabrÃ¼ck”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014

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