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category_of_open_sets [2014/10/28 22:24]
nikolaj
category_of_open_sets [2014/10/29 13:39] (current)
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 ===
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