Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most famous among these are the pointwise convergence, compact-open, and uniform topologies. Since the introduction of these topologies, however, a number of other function space topologies have been studied, including the fine and graph topologies. We now recall their definitions (including that of the uniform topology for context):
- The uniform topology is defined as the topology generated by the basis $\{B_{d_Y}(f,\epsilon)\ |\ f\in\mathcal{C}(X,Y)\text{ and }\epsilon\in\mathbb{R}_{>0}\}$, where $$B_{d_Y}(f,\epsilon)\overset{\mathrm{def}}{=}\{g\in\mathcal{C}(X,Y)\ |\ d_Y(f(x),g(x))<\epsilon\}.$$
- The fine topology is defined as the topology generated by the basis $\{B_{d_Y}(f,\epsilon)\ |\ f\in\mathcal{C}(X,Y)\text{ and }\epsilon\in\mathcal{C}(X,\mathbb{R}_{>0})\}$, where $$B_{d_Y}(f,\epsilon)\overset{\mathrm{def}}{=}\{g\in\mathcal{C}(X,Y)\ |\ \text{for each $x\in X$, we have $d_Y(f(x),g(x))<\epsilon(x)$}\}.$$ Note that here we have allowed $\epsilon$ to be a function from $X$ to $\mathbb{R}_{>0}$, instead of just a constant real number.
- The graph topology is defined as the topology generated by the basis $\{B_{d_Y}(f,\ell)\}_{(f,\ell)\in I}$, where:
- $f$ is a continuous function from $X$ to $Y$;
- $\ell$ is a lower semicontinuous function from $X$ to $\mathbb{R}_{>0}$;
- $B_{d_Y}(f,\epsilon)$ is given by $$B_{d_Y}(f,\epsilon)\overset{\mathrm{def}}{=}\{g\in\mathcal{C}(X,Y)\ |\ \text{for each $x\in X$, we have $d_Y(f(x),g(x))<\epsilon(x)$}\}.$$
These topologies are all comparable, and satisfy the following relations: $$\mathcal{T}_{\text{pointwise convergence}}\subset\mathcal{T}_{\text{compact-open}}\subset\mathcal{T}_{\text{uniform}}\subset\mathcal{T}_{\text{fine}}\subset\mathcal{T}_{\text{graph}}.$$
Now, the graph topology can in fact be (and was originally) defined by a different basis, which works for $Y$ an arbitrary topological space, given by $$\{\langle U\rangle\ |\ U\in\mathrm{Open}(X\times Y)\}$$ with $$\langle U\rangle\overset{\mathrm{def}}{=}\{f\in\mathcal{C}(X,Y)\ |\ \mathrm{Gr}(f)\subset U\},$$ where $\mathrm{Gr}(f)\subset X\times Y$ denotes the graph of $f$. One can also replace $\mathcal{C}(X,Y)$ in this definition with the set $Y^X$ of all maps from $X$ to $Y$ (continuous or not) to obtain a topology on $Y^X$.
The question, finally: Specialising to the case where $X$ is an arbitrary topological space and $Y$ is the set $\{0,1\}$ topologised with either the discrete, Sierpiński, or indiscrete topologies, we may consider the graph topology on the set $2^X\cong\mathcal{P}(X)$. Is it known how the resulting three graph topologies compare with the other commonly used powerset topologies for $\mathcal{P}(X)$, like the Vietoris topology?
(Incidentally, one result I know in this direction is the one mentioned in the introduction of the book Function Spaces with Uniform, Fine and Graph Topologies by McCoy–Kundu–Jindal, which states that the subspace topology on $\mathcal{C}(X,\mathbb{R})$ induced by the Vietoris topology on $\mathrm{Cld}(X\times\mathbb{R})$ agrees with the graph topology on $\mathcal{C}(X,\mathbb{R})$.)
