Condensations with extra properties
István Juhász, Jan van Mill, Lajos Soukup
TL;DR
The paper investigates condensations between infinite Tychonoff spaces, showing that locally compact spaces can condense onto separable spaces without admitting a compact separable condensation, and that for every cardinal $\kappa$ there exists a locally compact topological group of size $2^\kappa$ that condenses onto a compact space but not onto a compact topological group. It provides constructive positive results for condensing topological sums into separable targets via embeddings into Hilbert cubes, and then develops negative results using set-theoretic consistency (CH and forcing) to produce discrete counterexamples and nondiscrete obstructions, including that $\mathbb{R}$ condensing onto a compact space cannot yield a compact homogeneous target. The work also demonstrates obstructions in the group setting, such as that a condensation onto a compact homogeneous space can force a nonexistence result for certain groups, thereby answering questions of Arhangel$'$skii and Buzyakova in the negative and clarifying the role of cardinal arithmetic and independence results in condensation phenomena.
Abstract
We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $κ$ there is a locally compact topological group of cardinality $2^κ$ that can be condensed onto a compact space but not onto a compact topological group. These answer some questions of Arhangel'skii and Buzyakova.