CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological space is sequential if (in a certain sense) you can do topology in it using only sequences instead of more general nets.
Sequential spaces are a kind of nice topological space.
A sequential topological space is a topological space $X$ such that a subset $A$ of $X$ is closed if (hence iff) it contains all the limit points of all sequences of points of $A$—or equivalently, such that $A$ is open if (hence iff) any sequence converging to a point of $A$ must eventually be in $A$.
Equivalently, a topological space is sequential iff it is a quotient space (in $Top$) of a metric space.
Every Frechet–Uryson space is a sequential space.
Every topological space satisfying the first countability axiom is Frechet–Uryson, hence a sequential space. In particular, this includes any metrizable space.
Every quotient of a sequential space is sequential. In particular, every CW complex is also a sequential space. (Conversely, every sequential space is a quotient of a metrizable space, giving the alternative definition).
The category of sequential spaces is a coreflective subcategory of the category of all topological spaces.
The category of sequential spaces is a reflective subcategory of the category of subsequential spaces, much as $Top$ itself is a reflective subcategory of the category of all pseudotopological spaces.
The category of sequential spaces is cartesian closed. See also convenient category of topological spaces.