CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A -generated space is a topological space whose topology is the final topology induced by all maps , where runs over all the standard simplices.
The category of -generated spaces is coreflective in Top. It is also locally presentable, and supports a model structure. Thus, it is a nice category of spaces.
-generated spaces were originally proposed by Jeff Smith as a nice category of spaces for homotopy theory. A proof that they are locally presentable is in: