Remainders of metrizable spaces and a generalization of Lindelöf $\mit\Sigma$-spaces

A. V. Arhangel'skii¹

¹ Moscow 121165, Russia

Fundamenta Mathematicae, Tome 215 (2011) no. 1, pp. 87-100

Résumé

We establish some new properties of remainders of metrizable spaces. In particular, we show that if the weight of a metrizable space $X$ does not exceed $2^\omega $, then any remainder of $X$ in a Hausdorff compactification is a Lindelöf ${\mit \Sigma } $-space. An example of a metrizable space whose remainder in some compactification is not a Lindelöf ${\mit \Sigma } $-space is given. A new class of topological spaces naturally extending the class of Lindelöf ${\mit \Sigma } $-spaces is introduced and studied. This leads to the following theorem: if a metrizable space $X$ has a remainder $Y$ with a $G_\delta $-diagonal, then both $X$ and $Y$ are separable and metrizable. Some new results on remainders of topological groups are also established.

Détail

DOI : 10.4064/fm215-1-5

Keywords: establish properties remainders metrizable spaces particular weight metrizable space does exceed omega remainder hausdorff compactification lindel mit sigma space example metrizable space whose remainder compactification lindel mit sigma space given class topological spaces naturally extending class lindel mit sigma spaces introduced studied leads following theorem metrizable space has remainder delta diagonal separable metrizable results remainders topological groups established