CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Top is the category of topological spaces and continuous maps between them.
How exactly this is understood depends a bit on context: of course $Top$ forms an ordinary category. But it is also naturally an (∞,1)-category. This, in turn, may be presented by regarding $Top$ as a model category equipped with the Quillen model structure.
Moreover, what exactly counts as an object in $Top$ often varies in different contexts. For many applications it is useful to restrict to a subcategory of nice topological spaces such as compactly generated spaces or CW-complexes. There other other convenient categories of topological spaces.
The homotopy category of $Top$ with respect to weak homotopy equivalences is Ho(Top). This is the central object of study in homotopy theory. Regarded as an (∞,1)-category $Top$ is the archetypical homotopy theory, equivalent to ∞Grpd.
An axiomatic desciption of $Top$ building along the lines of ETCS for Set is discussed in