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metric_space [2014/10/25 16:02] nikolaj |
metric_space [2015/01/21 19:41] nikolaj |
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===== Metric space ===== | ===== Metric space ===== | ||
==== Set ==== | ==== 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]] |