CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
typical contexts
The forgetful functor $\Gamma : Top \to Set$ from Top to Set that sends any topological space to its underlying set has a left adjoint $Disc : Set \to Top$ and a right adjoint $Codisc : Set \to Top$.
For $S \in Set$
$Disc(S)$ is the topological space on $S$ in which every subset is an open set
this is called the discrete topology on $S$, $Disc(S)$ is called a discrete space;
$Codisc(S)$ is the topological space on $S$ whose only open sets are the empty set and $S$ itself
this is called the codiscrete topology on $S$ (also indiscrete topology or trivial topology), $Codisc(S)$ is called a codiscrete space .
For an axiomatization of this situation see codiscrete object.