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Let $Y$ be a locally compact $\sigma$-compact Hausdorff space and $q:Y\to X$ a quotient map with $X$ Hausdorff. Consider the following properties.

I. There is a dense subset $D\subseteq Y$ such that the restriction of $q$ to $D$ is injective.

II. The interior of every fibre (i.e. set of the form $q^{-1}(x)$, $x\in X$) in $Y$ is either empty or a singleton.

Then clearly I.$\Rightarrow$II. Suppose that $Y$ is a separable metric space satisfying II. Let $D_0$ be the (countable) set of isolated points in $Y$ and fix a countable base $\{U_i\}_{i\ge 1}$ for $Y\setminus \overline{D_0}$. Inductively chose a point from each $U_i$ which does not belong to any fibre with non-empty interior or any fibre previously chosen. This is possible because $U_i$ contains no isolated point and hence is an uncountable Baire space, and every fibre intersects $U_i$ in a closed set with empty interior. Let $D_1$ be the set thus obtained, and set $D=D_0\cup D_1$. Then $D$ is dense in $Y$ and $q$ restricted to $D$ is injective. So for locally compact separable metric spaces, II.$\Rightarrow$I.

Question: Does II.$\Rightarrow$I. for all locally compact $\sigma$-compact Hausdorff spaces?

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We have II and not I when $X$ is any non-empty locally compact $\sigma$-compact Hausdorff space without isolated points, $Z$ is any compact Hausdorff space such that the density of $Z$ is bigger than $|X|$, $Y=X\times Z$, and $q:X\times Z\to X$ is the projection map.

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    $\begingroup$ Very good! Many thanks once again. $\endgroup$
    – user558840
    Oct 3, 2020 at 7:19

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