Continuous pseudo-hairy spaces and continuous pseudo-fans

Janusz R. Prajs¹

¹ Institute of Mathematics Opole University Oleska 48 45-052 Opole, Poland Current address Department of Mathematics Idaho State University Pocatello, ID 83209, U.S.A.

Fundamenta Mathematicae, Tome 171 (2002) no. 2, pp. 101-116.

Résumé

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 $X$.

Détail

DOI : 10.4064/fm171-2-1

Keywords: compact metric space widetilde said continuous pseudo hairy space compact space subset widetilde provided there exists monotone retraction widetilde buildrel longrightarrow fibers pseudo arcs continuum widetilde joining different fibers intersects nbsp continuum called continuous pseudo fan compactum there point family cal pseudo arcs bigcup cal subcontinuum intersecting different elements cal contains cal homeomorphic respect hausdorff metric proved each compact metric space there exist continuous pseudo hairy space continuous pseudo fan nbsp