Some remarks concerning the cardinal min{r,d}

Prelegent(ci)

Piotr Zakrzewski

Afiliacja

University of Warsaw

Język referatu

polski

Termin

4 marca 2026 16:15

Pokój

p. 4050

Seminarium

Seminarium „Topologia i teoria mnogości”

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The talk is closely related to the results from a joint
article with Roman Pol and Lyubomyr Zdomskyy (see arXiv:2502.20887). In
that paper we introduced a cardinal d^* as the smallest κ such that any
two disjoint countable dense
sets in the Cantor set can be separated by sets each of which is an
intersection of at most κ-many open sets. We proved that d^*=min{r,d},
where d and r are the dominating and the reaping number, respectively.

In the first part of the talk I will sketch a direct proof that
d^*=bidi, the cardinal characteristic of the continuum studied earlier
by Szewczak and Tsaban. By a joint result of Szewczak, Tsaban and Meija,
bidi is also equal to min{r,d}.

In the second part of the talk, based on joint results with Roman PoI, I will point out how the topological
characterization od min{r,d}, expressed by the equality d^*=min{r,d},
can be used to obtain, under the assumption that d is not bigger than r,
examples of Menger sets of reals X and Y such that  the product spaces X
x Y and X x X are not Menger. Such examples were earlier given by
Szewczak, Tsaban and Zdomskyy.

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