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

Volume 215 / 2011

A. V. Arhangel'skii Fundamenta Mathematicae 215 (2011), 87-100 MSC: Primary 54A25; Secondary 54B05. DOI: 10.4064/fm215-1-5


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.


  • A. V. Arhangel'skiiMoscow 121165, Russia

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