## Continuous pseudo-hairy spaces and continuous pseudo-fans

### Volume 171 / 2002

#### Abstract

A compact metric space $\widetilde{X} $ is said to be
a *continuous pseudo-hairy space over* a compact space
$X\subset \widetilde{X} $ provided there exists an open, monotone retraction
$r: \widetilde{X} \buildrel {\rm onto}\over\longrightarrow
X $
such that all fibers $r^{-1}(x)$ are
pseudo-arcs and any continuum in $\widetilde{X}$
joining two different fibers of $r$ intersects $X$.
A continuum $Y_{X}$ is called a {\it continuous pseudo-fan of}
a compactum $X$ if there are a point $c\in Y_{X}$ and a
family ${\cal F}$ of pseudo-arcs such that $\bigcup {\cal F} = Y_{X} $,
any subcontinuum of $Y_{X}$ intersecting two different elements of
${\cal F}$ contains $c$, and
${\cal F}$ is homeomorphic to $X$ (with respect to the Hausdorff metric).
It is proved that for each compact metric space $X$ there exist
a continuous pseudo-hairy space over $X$ and a continuous pseudo-fan of $X$.