When a continuous map from a quotient space satisfies the sheaf condition?

Question

In the category of topological spaces, let E -> E/R -> X, where R is an equivalence relation.

Say E -> X satisfies the condition (*) iff every point in E has a open nbd such that it homeomorphic to its image which is open in X.

Question: if E -> X satisfies *, when does E/R -> X also satisfy * ?

My reading says it is exactly when R is open in E x E, but I can't prove it.

Answer 1

Actually, $R$ is open in $E \times E$ if and only if $E / R$ is a discrete topological space, so that condition is far too strong.

Recall that a local homeomorphism is a continuous map that satisfies your condition ($\ast$). First, an observation: if $g \circ f$ and $g$ are both local homeomorphisms, then so is $f$. Thus, $E / R \to X$ is a local homeomorphism if and only if $E \to E / R$ is a local homeomorphism, since you have assumed that $E \to X$ is a local homeomorphism. We have the following result:

$E \to E / R$ is a local homeomorphism if and only if the projections $R \to E$ are open maps and the diagonal $\Delta_E$ is an open subspace of $R$.

This is a straightforward exercise – but note that you will need to use the explicit description of the product topology on $E \times E$.