# A possible characterization of stratifiable spaces?

Let us recall that a regular topological space is semi-stratifiable if each point $$x\in X$$ has a countable family of neighborhoods $$(U_n(x))_{n\in\omega}$$ such that each closed subset $$F\subset X$$ is equal to the intersection $$\bigcap_{n\in\omega}U_n[F]$$ of its neighborhoods $$U_n[F]=\bigcup_{x\in F}U_n(x)$$.

If, moreover, $$F=\bigcap_{n\in\omega}\overline{U_n[F]}$$, then the space $$X$$ is called stratifiable.

Basic information on stratifiable spaces can be found in this survey paper of Gruenhage.

Definition. A topological space $$X$$ is called an

$$\bullet$$ $$r'$$-space its set $$X'$$ of non-isolated point is a retract of $$X$$;

$$\bullet$$ hereditarily $$r'$$-space if each subspace of $$X$$ is an $$r'$$-space.

Observe that each separable $$r'$$-space has separable set $$X'$$ of non-isolated points.

It can be shown that each stratifiable space is a (hereditary) $$r'$$-space.

On the other hand, any uncountable Mrowka-Isbell space $$X$$ is semi-stratifiable but fails to be an $$r'$$-space (since it is separable but its set $$X'$$ of non-isolated points is not separable).

Question. Is each semi-stratifiable hereditary $$r'$$-space $$X$$ stratifiable?

What will be the answer if we additionally assume that $$X$$ is hereditarily paracompact?

Added in Edit. The hereditary $$r'$$-space property indeed characterizes the stratifiability at least in the class of scattered spaces of finite scattered height, as shown by the following theorem (that can proved by induction):

Theorem. For a scattered space $$X$$ of finite scattered height $$\hbar(X)$$ the following conditions are equivalent:

1) $$X$$ is stratifiable;

2) each non-empty closed subset of $$X$$ is a $$G_\delta$$-set in $$X$$ and is a retract of $$X$$;

3) $$X$$ is hereditarily $$r'$$-space and each closed subset of $$X$$ is of type $$G_\delta$$;

4) for every $$n<\hbar(X)$$ the $$n$$-th derived set $$X^{(n)}$$ is a $$G_\delta$$-set in $$X$$ and $$X^{(n)}$$ is a retract of $$X$$.

The derived sets $$X^{(\alpha)}$$ of a topological space $$X$$ are defined for any ordinal $$\alpha$$ by the recursive formulas:

$$\bullet$$ $$X^{(0)}=X$$,

$$\bullet$$ $$X^{(1)}$$ is the set of non-isolated points in $$X$$, and

$$\bullet$$ $$X^{(\alpha)}=\bigcap_{\beta<\alpha}(X^{(\beta)})^{(1)}$$ for any ordinal $$\alpha>0$$.

The scattered height $$\hbar (X)$$ of a topological space $$X$$ is the smallest ordinal $$\alpha$$ such that $$X^{(\alpha+1)}=X^{(\alpha)}$$.

A topological space $$X$$ is scattered if $$X^{(\hbar(X))}=\emptyset$$.