CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A natural topology on mapping spaces of continuous functions.
Let $X$ and $Y$ be topological spaces. The set $Map(X,Y)$ (often denoted also $C(X,Y)$) of continuous maps from $X$ to $Y$ has a natural topology called the compact-open topology: a subbase of that topology consists of sets of the form $U_{K,V}$, where $K\subset X$ is compact and $V\subset Y$ is open?, which consists of all continuous maps $f:X\to Y$ such that $f(K)\subset V$.
If $Y$ is a metric space then the compact-open topology is the topology of uniform convergence on compact subsets in the sense that $f_n \to f$ in $Map(X,Y)$ with the compact-open topology iff for every compact subset $K\subset X$, $f_n \to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the topology of uniform convergence.
The compact-open topology is most sensible when the topology of $X$ is locally compact Hausdorff, for in this case $Map(X,Y)$ with the compact-open topology is an exponential object $Y^X$ in the category Top of all topological spaces. This implies the exponential law for spaces , i.e. the adjunction map is a bijection $Top(X,Map(Y,Z))\cong Top(X\times Y,Z)$ whenever $Y$ is locally compact Hausdorff; and it becomes a homeomorphism $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ if in addition $X$ is also Hausdorff. See also convenient category of topological spaces.
Ralph H. Fox, On Topologies for Function Spaces , Bull. AMS 51 (1945) pp.429-432. (pdf)