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category_of_open_sets [2014/10/28 22:24] nikolaj |
category_of_open_sets [2014/10/29 13:39] nikolaj |
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| | @#AAFFAA: inclusion | @#AAFFAA: $\mathrm{Op}(X)$ ... category | | | @#AAFFAA: inclusion | @#AAFFAA: $\mathrm{Op}(X)$ ... category | | ||
| | @#FF9944: definition | @#FF9944: $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$ | | | @#FF9944: definition | @#FF9944: $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$ | | ||
| - | | @#FFFDDD: for all | @#FFFDDD: $U,V\in\mathrm{Ob}_{\mathrm{Op}(X)}$ | | + | | @#FFFDDD: for all | @#FFFDDD: $V,U\in\mathrm{Ob}_{\mathrm{Op}(X)}$ | |
| - | | @#FF9944: definition | @#FF9944: $\mathrm{Op}(X)[U,V]\equiv\{i:U\to V\ |\ i(x)=x\}$ | | + | | @#FF9944: definition | @#FF9944: $\mathrm{Op}(X)[V,U]\equiv\{i:V\to U\ |\ i(x)=x\}$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | In the category of open sets, if one object $U$ is subset of another $V$, i.e. $U\subseteq V$, then the hom-set $\mathrm{Op}(X)[U,V]=\{i\}$ contains the only inclusion function and is empty otherwise. | + | In the category of open sets, the arrows are the inclusion functions. In the case $V\subseteq U$, the hom-set $\mathrm{Op}(X)[U,V]$ is the singleton $\{i\}$ and otherwise it's empty. |
| === Reference === | === Reference === | ||
