Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
topological_space [2014/10/25 16:48] nikolaj |
topological_space [2015/02/26 20:24] (current) nikolaj |
||
|---|---|---|---|
| Line 7: | Line 7: | ||
| | @#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]] | ||
