CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A simplicial topological space is a simplicial object in Top.
Often this is called just a simplicial space , if the context is clear.
A special case is that of simplicial manifolds.
Often one is interested in simplicial topological spaces with extra nice properties. See nice simplicial topological space for more on that.
As with simplicial objects in general, simplicial spaces may serve to model internal ∞-groupoids in Top. Notably there is a rich theory of simplicial topological groups.
simplicial topological space, nice simplicial topological space
A standard textbook reference is chapter 11 of