CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A $\Delta$-generated space is a topological space $X$ whose topology is the final topology induced by all maps $\Delta^n \to X$, where $\Delta^n$ runs over all the standard simplices.
The category of $\Delta$-generated spaces is coreflective in Top. It is also locally presentable, and supports a model structure. Thus, it is a nice category of spaces.
$\Delta$-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:
See also at directed homotopy theory.
Other references include:
Tadayuki Haraguchi: On model structure for coreflective subcategories of a model category (2013-04-12T12:42:18Z): arXiv:1304.3622v1, MR3289294, Zbl 1311.55027.
K. Shimakawa, K. Yoshida, T. Haraguchi: Homology and cohomology via enriched bifunctors (2010-10-16T10:31:57Z): arXiv:1010.3336v1.