Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is exponentiable. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equivalent to the seemingly weaker condition that for the Sierpinski space $Z = \mathbf{2}$ in particular (where the underlying set of $C(X, \mathbf{2})$ is in natural bijection with the topology of $X$), there is a topology so that $C(X, \mathbf{2})$ represents the functor $\hom_{\text{Top}}(X \times -, \mathbf{2})$: there is a natural isomorphism

$$\hom_{\text{Top}}(-, C(X, \mathbf{2})) \cong \hom_{\text{Top}}(X \times -, \mathbf{2}).$$

(Of course the topology on $C(X, \mathbf{2}) \cong \text{Open}(X)$ is uniquely determined by this universal property.)

Topologically, such $X$ are exactly core-compact spaces. To define this notion, 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 description is synonymous with saying: the topology $\text{Open}(X)$ is a continuous lattice, for which there is also a lot of literature, especially in so-called domain theory (after Dana Scott).

Core-compact spaces are a very mild generalization of locally compact spaces (defined here to mean spaces where the neighborhood filter at any point is generated by compact neighborhoods). All locally compact spaces are core-compact. If $X$ is Hausdorff or even merely sober, then $X$ is core-compact iff it is locally compact (along the lines Simon was suggesting, if we consider spatial locales), according to Theorem 8.3.10 in Non-Hausdorff Topology and Domain Theory by Goubault-Larrecq (partially viewable in Google Books).

It seems to be surprisingly hard to cook up (or even find in the literature!) a core-compact space that is not locally compact, but you can find such an example in this paper by Hofmann and Lawson (section 7).

Another fine compendium of results is Topologies on Spaces of Continuous Functions by Escardó and Heckmann (I believe it addresses aspects of your other recent question on admissible topologies on function spaces, but I've not had time to look into this carefully.)

As for 1), I'm currently somewhat skeptical that a nice answer can be given, but here again I've not had time to investigate this properly.

Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is exponentiable. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equivalent to the seemingly weaker condition that for the Sierpinski space $Z = \mathbf{2}$ in particular (where the underlying set of $C(X, \mathbf{2})$ is in natural bijection with the topology of $X$), there is a topology so that $C(X, \mathbf{2})$ represents the functor $\hom_{\text{Top}}(X \times -, \mathbf{2})$: there is a natural isomorphism

$$\hom_{\text{Top}}(-, C(X, \mathbf{2})) \cong \hom_{\text{Top}}(X \times -, \mathbf{2}).$$

(Of course the topology on $C(X, \mathbf{2}) \cong \text{Open}(X)$ is uniquely determined by this universal property.)

Topologically, such $X$ are exactly core-compact spaces. To define this notion, 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 description is synonymous with saying: the topology $\text{Open}(X)$ is a continuous lattice, for which there is also a lot of literature, especially in so-called domain theory (after Dana Scott).

Core-compact spaces are a very mild generalization of locally compact spaces (defined here to mean spaces where the neighborhood filter at any point is generated by compact neighborhoods). All locally compact spaces are core-compact. If $X$ is Hausdorff or even merely sober, then $X$ is core-compact iff it is locally compact (along the lines Simon was suggesting, if we consider spatial locales), according to Theorem 8.3.10 in Non-Hausdorff Topology and Domain Theory by Goubault-Larrecq (partially viewable in Google Books).

It seems to be surprisingly hard to cook up (or even find in the literature!) a core-compact space that is not locally compact, but you can find such an example in this paper by Hofmann and Lawson (section 7).

Another fine compendium of results is Topologies on Spaces of Continuous Functions by Escardó and Heckmann (I believe it addresses aspects of your other recent question on admissible topologies on function spaces, but I've not had time to look into this carefully.)

As for 1), I'm currently somewhat skeptical that a nice answer can be given, but here again I've not had time to investigate this properly.

Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is exponentiable. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equivalent to the seemingly weaker condition that for the Sierpinski space $Z = \mathbf{2}$ in particular (where the underlying set of $C(X, \mathbf{2})$ is in natural bijection with the topology of $X$), there is a topology so that $C(X, \mathbf{2})$ represents the functor $\hom_{\text{Top}}(X \times -, \mathbf{2})$: there is a natural isomorphism

$$\hom_{\text{Top}}(-, C(X, \mathbf{2})) \cong \hom_{\text{Top}}(X \times -, \mathbf{2}).$$

(Of course the topology on $C(X, \mathbf{2}) \cong \text{Open}(X)$ is uniquely determined by this universal property.)

Topologically, such $X$ are exactly core-compact spaces. To define this notion, 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 description is synonymous with saying: the topology $\text{Open}(X)$ is a continuous lattice, for which there is also a lot of literature, especially in so-called domain theory (after Dana Scott).

Core-compact spaces are a very mild generalization of locally compact spaces (defined here to mean spaces where the neighborhood filter at any point is generated by compact neighborhoods). All locally compact spaces are core-compact. If $X$ is Hausdorff or even merely sober, then $X$ is core-compact iff it is locally compact (along the lines Simon was suggesting, if we consider spatial locales), according to Theorem 8.3.10 in Non-Hausdorff Topology and Domain Theory by Goubault-Larrecq (partially viewable in Google Books).

It seems to be surprisingly hard to cook up (or even find in the literature!) a core-compact space that is not locally compact, but you can find such an example in this paper by Hoffmann and Lawson (section 7).

Another fine compendium of results is Topologies on Spaces of Continuous Functions by Escardó and Heckmann (I believe it addresses aspects of your other recent question on admissible topologies on function spaces, but I've not had time to look into this carefully.)

As for 1), I'm currently somewhat skeptical that a nice answer can be given, but here again I've not had time to investigate this properly.

Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is exponentiable. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equivalent to the seemingly weaker condition that for the Sierpinski space $Z = \mathbf{2}$ in particular (where the underlying set of $C(X, \mathbf{2})$ is in natural bijection with the topology of $X$), there is a topology so that $C(X, \mathbf{2})$ represents the functor $\hom_{\text{Top}}(X \times -, \mathbf{2})$: there is a natural isomorphism

$$\hom_{\text{Top}}(-, C(X, \mathbf{2})) \cong \hom_{\text{Top}}(X \times -, \mathbf{2}).$$

(Of course the topology on $C(X, \mathbf{2}) \cong \text{Open}(X)$ is uniquely determined by this universal property.)

Topologically, such $X$ are exactly core-compact spaces. To define this notion, 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 description is synonymous with saying: the topology $\text{Open}(X)$ is a continuous lattice, for which there is also a lot of literature, especially in so-called domain theory (after Dana Scott).

Core-compact spaces are a very mild generalization of locally compact spaces (defined here to mean spaces where the neighborhood filter at any point is generated by compact neighborhoods). All locally compact spaces are core-compact. If $X$ is Hausdorff or even merely sober, then $X$ is core-compact iff it is locally compact (along the lines Simon was suggesting, if we consider spatial locales), according to Theorem 8.3.10 in Non-Hausdorff Topology and Domain Theory by Goubault-Larrecq (partially viewable in Google Books).

It seems to be surprisingly hard to cook up (or even find in the literature!) a core-compact space that is not locally compact, but you can find such an example in this paper by Hofmann and Lawson (section 7).

Another fine compendium of results is Topologies on Spaces of Continuous Functions by Escardó and Heckmann (I believe it addresses aspects of your other recent question on admissible topologies on function spaces, but I've not had time to look into this carefully.)

As for 1), I'm currently somewhat skeptical that a nice answer can be given, but here again I've not had time to investigate this properly.