## Abstract

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

03E55, 54A25

Original language | English |
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Journal | Unknown Journal |

Publication status | Published - 2017 Sep 22 |

## Keywords

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

## ASJC Scopus subject areas

- General