Open Subsets of LF-spaces

Kotaro Mine¹ ; Katsuro Sakai¹

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

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 1, pp. 25-37.

Résumé

Let $F=\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{i}\mathrm{m}{F}_n$ be an infinite-dimensional LF-space with density $\mathrm{dens}F=\tau$ ($\ge {\mathrm{\aleph }}_0$) such that some $F_n$ is infinite-dimensional and $\mathrm{dens}{F}_n=\tau$. It is proved that every open subset of $F$ is homeomorphic to the product of an ${\ell }_2(\tau )$-manifold and ${\mathbb{R}}^{\mathrm{\infty }}=\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{i}\mathrm{m}{\mathbb{R}}^n$ (hence the product of an open subset of ${\ell }_2(\tau )$ and ${\mathbb{R}}^{\mathrm{\infty }}$). As a consequence, any two open sets in $F$ are homeomorphic if they have the same homotopy type.

Détail

DOI : 10.4064/ba56-1-4

Mots-clés : mathop ind lim infinite dimensional lf space density mathop dens tau geq aleph infinite dimensional mathop dens tau proved every subset homeomorphic product ell tau manifold mathbb infty mathop ind lim mathbb hence product subset ell tau mathbb infty consequence sets homeomorphic have homotopy type