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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?

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Fix a graph $G=(V,E)$ where $E \subset [V]^2$.

Give the discrete topology to both $X:=V \cup E \cup \{\infty\}$ and $Y:=2$. For each $v \in V$ let $f_v:X \to Y$ be the (automatically continuous) map defined by $f_v^{-1}(1)=\{v\}\cup\{e\in E: v\in e\}$.

The graph induced in $\{f_v :v\in V\}\subseteq \mathrm{Hom}(X,Y)$ by the similarity relation is isomorphic to $G$: for $u \neq v$, the map $f_u$ is similar to the map $f_v$ if and only if $f_u^{-1}(1) \cap f_v^{-1}(1) \neq \emptyset$ if and only if $\{u,v\}\in E$. Note that $\infty$ is always in $f_u^{-1}(0) \cap f_v^{-1}(0)$, so that this intersection is nonempty for free.

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