TITLE: What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?
QUESTION [9 upvotes]: For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, then $\times X$ has a right adjoint, namely $Y\mapsto Y^X$(equipped with compact-open topology). I also heard that the right adjoint of $\times X$ doesn't exist for every topological space $X$. So I wonder is there any more visible description of the class of topological spaces $X$ such that the left adjoints of$\times X$ exist? Any idea will be appreciated.

REPLY [4 votes]: In answer to a question that came up in comments under Najib's answer, let me point out that the category of pseudotopological spaces is a locally small cartesian closed category that contains $\text{Top}$ as a full subcategory. Actually, this category $\text{PsTop}$ has the even stronger property of being complete and locally cartesian closed and indeed is a quasitopos. 
Exactly the same remarks apply to the even larger category of convergence spaces (larger in the sense that pseudotopological spaces are special types of convergence spaces). 
Moreover, we have the very nice fact that the inclusion functor $i: \text{Top} \hookrightarrow \text{PsTop}$ preserves exponential objects, in the sense that if $X, Y$ are topological spaces such that the exponential $Y^X$ exists in $\text{Top}$, then the canonical map $i(Y^X) \to i(Y)^{i(X)}$ is an isomorphism (of course, the exponential on the right is formed qua the cartesian closed structure on $\text{PsTop}$). 
One way of thinking about this is that for any topological spaces $Y, X$, the natural candidate for an exponential topology on the set of continuous functions $\hom_{\text{Top}}(X, Y)$ (if one exists) is always given by continuous convergence, meaning the topology that is naturally associated with the convergence relation that defines the exponential in $\text{PsTop}$. That is, a topology (called the natural topology on $\hom_{\text{Top}}(X, Y)$) can always be formed that corresponds to continuous convergence, and so must be the exponential topology on $Y^X$ whenever there is one, and it is always the case that this natural topology is the finest topology on $\hom_{\text{Top}}(X, Y)$ such that given any topological space $A$ and a continuous function $A \times X \to Y$, the function $A \to \hom_{\text{Top}}(X, Y)$ formed by currying is also continuous. For basics on continuous convergence, see these notes by Mike Shulman (and the references given therein).