CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Given a topological space $X$, the category of open subsets $Op(X)$ of $X$ is the category whose
objects are the open subsets $U \hookrightarrow X$ of $X$;
morphisms are the inclusions $\array{ V &&\hookrightarrow && U \\ & \searrow && \swarrow \\ && X }$ of open subsets into each other.
The category $Op(X)$ is a poset, in fact a frame (dually a locale): it is the frame of opens of $X$.
The category $Op(X)$ is naturally equipped with the structure of a site, where a collection $\{U_i \to U\}_i$ of morphisms is a cover precisely if their union in $X$ equals $U$:
The category of sheaves on $Op(X)$ equipped with this site structure is usually written