Dekui Peng

Cardinal Spectra.

Let P be a cardinal invariant of topological groups. We call the set

\{P(H) : H \text{ is a dense subgroup of } G\}

the P-spectrum of dense subgroups of G. We can also define the P-spectrum of quotient groups (subgroups, closed subgroups, etc.) in the obvious way.

This notion is inspired by the double density spectrum of a topological space defined by Juhász, van Mill, Soukup, and Szentmiklóssy.

We mainly discuss the density spectrum of dense subgroups and the weight spectrum of quotient groups in this talk. The main results are:

  • For every compact group G, the density spectrum of dense subgroups is the interval [d(G), w(G)].
  • For every subset E \subseteq [\omega, \mathfrak{c}] with \sup E \in E, there exists a precompact abelian group whose weight spectrum of quotient groups is E. Moreover, if \omega \in E, then the group can be chosen to be pseudocompact.

References

[1] I. Juhász, J. van Mill, L. Soukup, and Z. Szentmiklóssy, The double density spectrum of a topological space, Israel J. Math., 255 (2023), 383—400.

[2] D. Peng, Densities and Weights of Quotients of Precompact Abelian Groups, to appear in Israel J. Math., available at https://arxiv.org/abs/2211.06831v4.