CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
When in a convenient category of topological spaces, e.g. compactly generated spaces, the category is cartesian closed, so that there is an adjunction between the mapping space and the cartesian product in that category. For general topological spaces there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable.
For $C$ a category with finite products, recall that an object $c$ is exponentiable if the functor $c \times -: C \to C$ has a right adjoint, usually denoted $(-)^c: C \to C$.
Let Top be the category of all topological spaces. An object $X$ of $Top$ is exponentiable if and only if $X \times -: Top \to Top$ preserves coequalizers, or equivalently quotient spaces.
This functor always preserves coproducts, so this condition is equivalent to saying that $X \times -$ preserves all small colimits. This is then equivalent to exponentiability by the adjoint functor theorem.
This condition, however, is not really any more explicit. More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition.
For open subsets $U$ and $V$ of a topological space $X$, we write $V\ll U$ to mean that any open cover of $U$ admits a finite subcover of $V$; this is read as $V$ is relatively compact under $U$ or $V$ is way below $U$. We say that $X$ is core-compact if for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ with $V\ll U$. In other words, $X$ is core-compact iff for all open subsets $V$, we have $V = \bigcup \{ U | U\ll V \}$. This says essentially the same thing as saying that the open-set lattice of $X$ is a continuous lattice, which yields the corresponding definition for locales.
An object $X$ of $Top$ is exponentiable if and only if it is core-compact.
If $X$ is Hausdorff, then core-compactness is equivalent to local compactness; thus in particular all locally compact Hausdorff spaces are exponentiable. In fact, local compactness implies exponentiability even without the Hausdorff condition, if local compactness is defined appropriately (for every point the compact neighborhoods form a neighborhood basis). For this reason, that core-compactness is also called quasi local compactness.
When $X$ is core-compact, we can explicitly describe the exponential topology on $Y^X$ (whose points are continuous maps $f: X \to Y$). It is generated by subbasis elements $O_{U,V}$, for $U$ an open subset of $X$ and $V$ an open subset of $Y$, where a continuous map $f\colon X \to Y$ belongs to $O_{U,V}$ iff $U\ll f^{-1}(V)$:
If $X$ and $Y$ are Hausdorff, then this topology on $Y^X$ coincides with the compact-open topology.
Exponentiable (i.e. core-compact) spaces can also be characterized in terms of ultrafilter convergence. Recall that a topological space can equivalently be defined as a lax algebra for the ultrafilter monad $U$ on the (1,2)-category Rel of sets and relations. In other words, it consists of a set $X$ and a relation $R\colon U X \to X$ called “convergence”, such that $id_X \subseteq R \circ \eta$ and $R\circ U R \subseteq R\circ \mu$, where $\eta$ and $\mu$ are the unit and multiplication of the ultrafilter monad, regarded as relations. In the paper
it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law $R\circ U R = R\circ \mu$.
Some intuition for this characterization can be obtained as follows. Consider the standard non-locally-compact space, the rationals $\mathbb{Q}$ as a subspace of the reals $\mathbb{R}$. Suppose that $x$ is a rational number and that $y_n$ is a sequence of irrationals converging to $x$. Then for each $n$ we can find a sequence $z^n_m$ of rationals which converges to $y_n$; hence the $z^n_m$ form a “sequence of sequences” which “globally converges” to $x$ in $\mathbb{Q}$, i.e. which are related to $x$ by the composite relation $R\circ \mu$, but for which does not converge elementwise to an intermediate sequence which in turn converges to $x$, i.e. it is not related to $x$ by the relation $R \circ U R$. It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact.
If $X$ is exponentiable, then the exponential law gives us an isomorphism of sets $Map(Y,B^X) \cong Map(X\times Y,B)$ for any other spaces $B$ and $Y$. If $Y$ is also exponentiable, then the Yoneda lemma yields from this a homeomorphism $B^{X\times Y} \cong (B^X)^Y$. However, we can also say some things in general without all spaces involved being exponentiable.
We now agree to denote by $Map(X,Y)=X^Y$ the space of continuous maps $X\to Y$ in the compact-open topology.
Let $X,Y,B$ be topological spaces. For any $f\in B^{X\times Y}$, the formula
defines a continuous map $\theta f:Y\to B^X$ which we call the map adjoint to $f$, or the adjunct of $f$.
The adjunction map
is a one-to-one function, and if $X$ is locally compact and Hausdorff then $\theta$ is a bijection. Independently from that assumption on $X$, if $Y$ is Hausdorff, then $\theta$ is continuous in the compact-open topology
If both assumptions (on $X$ and $Y$) are satisfied, then $\theta$ is not only a continuous bijection, but also open, hence a homeomorphism.
There is also a version for based (= pointed) topological spaces. The cartesian product then needs to be replace by the smash product of the based spaces. Regarding that the maps preserve the base point, the adjunction map $\theta$ induces the adjunction map
where the mapping space $Map_*$ for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point.
It appears that $\theta_*$ is again one-to-one and continuous, and it is bijective if $X$ is locally compact Hausdorff. If $Y$ is also Hausdorff then $\theta_*$ is a homeomorphism.
M. Escardo and R. Heckmann, Topologies of spaces of continuous functions, 2001.
Eva Lowen-Colebunders and Günther Richter, An elementary approach to exponential spaces, MR.
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 1.3 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)