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topological_space [2014/10/25 16:48] nikolaj |
topological_space [2015/02/26 20:24] nikolaj |
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| @#55EE55: postulate | @#55EE55: $S$ ... finite $\Rightarrow \bigcap S\in \mathcal T$ | | | @#55EE55: postulate | @#55EE55: $S$ ... finite $\Rightarrow \bigcap S\in \mathcal T$ | | ||
- | ==== Discussion ==== | + | ----- |
We call $\mathcal T$ the topology and its elements the open (sub-)sets of $X$. | 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 $(-\tfrac{1}{n},\tfrac{1}{n})$. | 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 ==== | + | === Reference === |
Wikipedia: [[http://en.wikipedia.org/wiki/Topological_space|Topological space]] | Wikipedia: [[http://en.wikipedia.org/wiki/Topological_space|Topological space]] | ||
- | ==== Parents ==== | + | |
+ | ----- | ||
=== Requirements === | === Requirements === | ||
[[Arbitrary union]], [[Arbitrary intersection]] | [[Arbitrary union]], [[Arbitrary intersection]] |