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2011•arXiv (Cornell University)Open access

Weak extent, submetrizability and diagonal degrees

Désirée Basile, Angelo Bella, Guit-Jan Ridderbos

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Abstract

We show that if $X$ has a zero-set diagonal and $X^2$ has countable weak extent, then $X$ is submetrizable. This generalizes earlier results from Martin and Buzyakova. Furthermore we show that if $X$ has a regular $G_δ$-diagonal and $X^2$ has countable weak extent, then $X$ condenses onto a second countable Hausdorff space. We also prove several cardinality bounds involving various types of diagonal degree.

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We show that if $X$ has a zero-set diagonal and $X^2$ has countable weak extent, then $X$ is submetrizable. This generalizes earlier results from Martin and Buzyakova. Furthermore we show that if $X$ has a regular $G_δ$-diagonal and $X^2$ has countable weak extent, then $X$ condenses onto a second countable Hausdorff space. We also prove several cardinality bounds involving various types of diagonal degree.

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Available abstract

We show that if $X$ has a zero-set diagonal and $X^2$ has countable weak extent, then $X$ is submetrizable. This generalizes earlier results from Martin and Buzyakova. Furthermore we show that if $X$ has a regular $G_δ$-diagonal and $X^2$ has countable weak extent, then $X$ condenses onto a second countable Hausdorff space. We also prove several cardinality bounds involving various types of diagonal degree.

Key concepts: Diagonal, Mathematics, Mathematical economics, Combinatorics, Geometry

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