Tsukuba Journal of Mathematics

Characterizations and Properties of Stratifiable Spaces

Huaipeng Chen

Full-text: Open access


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 $(X,\mu)$.

Article information

Tsukuba J. Math., Volume 32, Number 2 (2008), 253-276.

First available in Project Euclid: 30 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Chen, Huaipeng. Characterizations and Properties of Stratifiable Spaces. Tsukuba J. Math. 32 (2008), no. 2, 253--276. doi:10.21099/tkbjm/1496165228. https://projecteuclid.org/euclid.tkbjm/1496165228

Export citation