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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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metric_space [2014/11/20 12:59] nikolaj |
metric_space [2015/01/21 19:41] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $X$ ... set | | | @#55CCEE: context | @#55CCEE: $X$ ... set | | ||
| | @#FFBB00: definiendum | @#FFBB00: $\langle X,d\rangle\in\mathrm{it}$ | | | @#FFBB00: definiendum | @#FFBB00: $\langle X,d\rangle\in\mathrm{it}$ | | ||
| - | | @#55EE55: postulate | @#55EE55: $d\in$ metric $X$ | | + | | @#55EE55: postulate | @#55EE55: $d$ ... metric $X$ | |
| - | ==== Discussion ==== | + | ----- |
| We can reconstruct the set underlying a metric via $\text{dom}(\text{dom}(d))=\text{dom}(X\times X)=X$, so the set of metrics and the set of metric spaces over $X$ are in bijection. | We can reconstruct the set underlying a metric via $\text{dom}(\text{dom}(d))=\text{dom}(X\times X)=X$, so the set of metrics and the set of metric spaces over $X$ are in bijection. | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Metric_space|Metric space]] | Wikipedia: [[http://en.wikipedia.org/wiki/Metric_space|Metric space]] | ||
| - | ==== Parents ==== | + | |
| - | === Refinement of === | + | ----- |
| - | [[Hausdorff space]] | + | |
| === Requirements* === | === Requirements* === | ||
| [[Metric]] | [[Metric]] | ||
| + | === Refinement of === | ||
| + | [[Hausdorff space]] | ||
| === Equivalent to === | === Equivalent to === | ||
| [[Metric]] | [[Metric]] | ||
