# Gδ-TOPOLOGY AND COMPACT CARDINALS

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## Abstract

For a topological space X, let Xδ be the space X with Gδ-topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of Xδ is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of Xδ is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with Gδ-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with Gδ-topology. For the square of a Lindelöf space, using weak Gδ-topology, we prove that the following are consistent: (1) the least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.

03E55, 54A25

Original language English Unknown Journal Published - 2017 Sep 22

## Keywords

• -strongly compact cardinal-strongly compact cardinal.
• And phrases. cardinal function
• G-topology
• Lindelöf degree

• General