$G_\delta$-topology and compact cardinals
Abstract
For a topological space $X$, let $X_\delta$ be the space $X$ with $G_\delta$-topology of $X$. For an uncountable cardinal $\kappa$, we prove that the following are equivalent: (1) $\kappa$ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with $G_\delta$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_\delta$-topology. For the square of a Lindelöf space, using weak $G_\delta$-topology, we prove that the following are consistent: (1) the least $\omega_1$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2017
- DOI:
- 10.48550/arXiv.1709.07991
- arXiv:
- arXiv:1709.07991
- Bibcode:
- 2017arXiv170907991U
- Keywords:
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- Mathematics - Logic;
- Mathematics - General Topology;
- 03E55;
- 54A25