Characterizations and Properties of Stratifiable Spaces

Abstract

In this paper, we prove some properties and characterizations of stratifiable spaces and the following theorem:

Theorem. The following are equivalent:

1. $(Y,\tau)$ is a stratifiable space.

2. There is a zero-dimension submetric stratifiable space $(X, \mu)$ with $M_{3}$-structures and an irreducible perfect map $f:(X,\mu) \to (Y,\tau)$.

A stratifiable space $(X,\mu)$ is said to have an $M_3$-structure if $(X,\mu)$ satisfies the following conditions $A$ and $B$:

$A$: There is a countable collection $\mathscr{H} = \cup \mathscr{H}_{n}$ of $\rho$-closed sets such that:

1. $H(n',i') \subset H(n,i)$ or $\rho(H(n,i), H(n',i')) = r \gt 0$ if $H(n,i), H(n',i') \in \mathscr{H}$ with $n' \gt n$.

2. $\mathscr{H}_{n}$ is a partition of $X$ for each $n \in \mathbb{N}$.

$B$: There is a $g$-function $\mathscr{W}$ such that:

1. $\cap_{n}W(n,x) = \{x\}$.

2. $x \in W(n,x_{n})$, then $\{x_{n}: n \in \mathbb{N} \}$ converges to $x$.

3. If $H$ is closed and $x \notin H$, $x \notin Cl_{\mu}\left(\cup\{W(n,x'): x' \in H\}\right)$ for some $n$.

4. $x' \in W(n,x)$ implies $W(n,x') \subset W(n,x)$.

5. $H(n,i) \cap \left(\cup \mathscr{W}_{nj} \right) = \emptyset$ if $j \gt i$.

6. $W(n,x) \subset W(n-1,x)$.

7. Each $\mathscr{W}_{nm}$ is a $\rho$-discrete $\rho$-clopen collection.

8. $W(n,x) \subset c(n,x) \in \mathscr{C}$ for each $x \in X.$

Here $\mathscr{C}$ is a $g$-function of the stratifiable space $(X,\mu)$.