It is known that if $X$ is a (metric) ANR, then $X$ is locally equiconnected, that is, there is a neighborhood $V$ of the diagonal $Delta X subseteq X times X$ and a continuous function $$f colon V times (0,1) rightarrow X$$
- For every $(x,y) in V$, the path $f(x,y,-) colon (0,1) rightarrow X$ starts at $x$ and ends at $y$.
- For every $x in X$, the path $f(x,x,-) colon (0,1) rightarrow X$ is the constant path at $x$.
(Side note: Local equiconnectivity is equivalent to the diagonal map $Delta colon X rightarrow X times X$ being a Hurewicz cofibration.)
Let us also assume that $X$ is compact. My question is: Can we choose the $U$ and $f$ such that when $x neq y$ in the 1st condition, the path connecting them is a simple path?
Remark: It follows from Lemma 2.1 of the paper “A remark on simple path fields in polyhedra of characteristic zero” by Fadell that the answer is yes when $X$ is a finite simplicial complex. I am interested in a (strict) generalization of this result.