===== Topological space =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\langle X,\mathcal T\rangle \in \mathrm{it} $ |
| @#55EE55: postulate   | @#55EE55: $X,\emptyset\in \mathcal T$ |
| @#FFFDDD: for all | @#FFFDDD: $S\subseteq \mathcal T$ |
| @#55EE55: postulate   | @#55EE55: $\bigcup S\in \mathcal T$ |
| @#55EE55: postulate   | @#55EE55: $S$ ... finite $\Rightarrow \bigcap S\in \mathcal T$ |

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We call $\mathcal T$ the topology and its elements the open (sub-)sets of $X$.

A comment on the intersection axiom requiring finiteness: A major motivation for topological spaces is $\mathbb R^n$ with the sets "open ball" and in this setting, an infinite intersection of open sets need not be open. E.g. consider the set of open intevals $(-\tfrac{1}{n},\tfrac{1}{n})$.

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Topological_space|Topological space]]

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=== Requirements ===
[[Arbitrary union]], [[Arbitrary intersection]]