# Dense subset on which quotient map is injective

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?

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.