Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are *similar* if for all $V\subseteq Y$ open we have either

- $f^{-1}(V) = g^{-1}(V) = \emptyset$, or
- $f^{-1}(V) \cap g^{-1}(V) \neq \emptyset$.

Note that this relation is reflexive and symmetric (but not necessarily transitive). Let $\text{Hom}(X,Y)$ denote the set of all continuous maps from $X$ to $Y$. We make $\text{Hom}(X,Y)$ into a graph by saying that $\{f,g\}\subseteq \text{Hom}(X,Y)$ form an edge iff $f\neq g$ and $f,g$ are similar.

Given any (finite or infinite) graph $G$, are there Hausdorff spaces $X,Y$ such that $\text{Hom}(X,Y)$ has an induced subgraph that is isomorphic to $G$? And has this graph structure on $\text{Hom}(X,Y)$ been studied before?