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Category of open sets
Set
context | $\langle X,\mathcal T\rangle$ … topological space |
inclusion | $\mathrm{Op}(X)$ … category |
definition | $\mathrm{Ob}_{\mathrm{Op}(X)}\equiv \mathcal T$ |
for all | $U,V\in\mathrm{Ob}_{\mathrm{Op}(X)}$ |
definition | $\mathrm{Op}(X)[U,V]\equiv\{i:U\to V\ |\ i(x)=x\}$ |
Discussion
In the category of open sets, If one object $U$ is subset of another $V$, the hom-set $\mathrm{Op}(X)[U,V]=\{i\}$ contains the only inclusion function and is empty otherwise.
Reference
nLab: Thin category