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 $X$.
