Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato ¹   ; Eiichi Matsuhashi ¹

¹ Institute of Mathematics University of Tsukuba Ibaraki, 305-8571 Japan

Colloquium Mathematicum, Tome 106 (2006) no. 1, pp. 83-91

Résumé

The first author has recently proved that if $f: X \to Y$ is a $k$-dimensional map between compacta and $Y$ is $p$-dimensional ($0 \le k,p \infty $), then for each $0 \leq i \leq p+k$, the set of maps $g$ in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f \times g:X \to Y\times I^{p+2k+1-i}$ is an $(i+1)$-to-$1$ map is a dense $G_{\delta }$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if $f$ : $X \to Y$ is as above and $D_{j}$ $(j=1,\dots ,k)$ are superdendrites, then the set of maps $h$ in $C(X,\prod _{j=1}^{k}D_{j}\times I^{p+1-i})$ such that $f \times h:X \to Y\times (\prod _{j=1}^{k}D_{j}\times I^{p+1-i})$ is $(i+1)$-to-$1$ is a dense $G_{\delta }$-subset of $C(X,\prod _{j=1}^{k}D_{j}\times I^{p+1-i})$ for each $0\leq i \leq p$.

Détail

DOI : 10.4064/cm106-1-7

Keywords: first author has recently proved k dimensional map between compacta p dimensional infty each leq leq set maps space i diagonal product times times i to map dense delta subset i paper prove above dots superdendrites set maps prod times i times times prod times i to dense delta subset prod times i each leq leq