Abstract
In this paper, we prove some properties and characterizations of stratifiable spaces and the following theorem:
Theorem. The following are equivalent:
$(Y,\tau)$ is a stratifiable space.
There is a zero-dimension submetric stratifiable space $(X, \mu)$ with $M_{3}$-structures and an irreducible perfect map $f:(X,\mu) \to (Y,\tau)$.
$A$: There is a countable collection $\mathscr{H} = \cup \mathscr{H}_{n}$ of $\rho$-closed sets such that:
$H(n',i') \subset H(n,i)$ or $\rho(H(n,i), H(n',i')) = r \gt 0$ if $H(n,i), H(n',i') \in \mathscr{H}$ with $n' \gt n$.
$\mathscr{H}_{n}$ is a partition of $X$ for each $n \in \mathbb{N}$.
$B$: There is a $g$-function $\mathscr{W}$ such that:
$\cap_{n}W(n,x) = \{x\}$.
$x \in W(n,x_{n})$, then $\{x_{n}: n \in \mathbb{N} \}$ converges to $x$.
If $H$ is closed and $x \notin H$, $x \notin Cl_{\mu}\left(\cup\{W(n,x'): x' \in H\}\right)$ for some $n$.
$x' \in W(n,x)$ implies $W(n,x') \subset W(n,x)$.
$H(n,i) \cap \left(\cup \mathscr{W}_{nj} \right) = \emptyset$ if $j \gt i$.
$W(n,x) \subset W(n-1,x)$.
Each $\mathscr{W}_{nm}$ is a $\rho$-discrete $\rho$-clopen collection.
$W(n,x) \subset c(n,x) \in \mathscr{C}$ for each $x \in X.$
Citation
Huaipeng Chen. "Characterizations and Properties of Stratifiable Spaces." Tsukuba J. Math. 32 (2) 253 - 276, December 2008. https://doi.org/10.21099/tkbjm/1496165228
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