Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

• Main Authors: Dmitri Shakhmatov, Víctor Hugo Yañez
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Axioms
• Subjects: pseudocompact; strongly pseudocompact; p-compact; selectively sequentially pseudocompact; pseudo-ω-bounded; non-trivial convergent sequence; separable; free precompact Boolean group; reflexive group; maximal space; ultrafilter space
• Online Access: https://www.mdpi.com/2075-1680/7/4/86

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group <i>G</i> without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter <i>p</i> on the set <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">N</mi> </semantics> </math> </inline-formula> of natural numbers and every sequence <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of non-empty open subsets of <i>G</i>, one can choose a point <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>∈</mo> <msub> <mi>U</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> </inline-formula> for all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula> in such a way that the resulting sequence <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> has a <i>p</i>-limit in <i>G</i>; that is, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>:</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>∈</mo> <mi>V</mi> <mo>}</mo> <mo>∈</mo> <mi>p</mi> </mrow> </semantics> </math> </inline-formula> for every neighbourhood <i>V</i> of <i>x</i> in <i>G</i>. In particular, <i>G</i> is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group <i>G</i> above is not pseudo-<inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>⨁</mo> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </semantics> </math> </inline-formula>, where each space <inline-formula> <math display="inline"> <semantics> <msub> <mi>X</mi> <mi>i</mi> </msub> </semantics> </math> </inline-formula> is either maximal or discrete, contains no infinite separable pseudocompact subsets.