CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Could not include topos theory - contents
Given a topological space $X$, the open subspaces of $X$ form a poset which is in fact a frame. This is the frame of open subspaces of $X$. When thought of as a locale, this is the topological locale $\Omega(X)$. When thought of as a category, this is the category of open subsets of $X$.
Similarly, given a locale $X$, the open subspaces of $X$ form a poset which is in fact a frame. This is the frame of open subspaces of $X$. When thought of as a locale, this is simply $X$ all over again. When thought of as a category, this is a site whose topos of sheaves is a localic topos.
The frame of open subsets of the point is given by the power set of a singleton, or more generally by the object of truth values of the ambient topos.