GAMES AND HEREDITARY BAIRENESS IN HYPERSPACES AND SPACES OF PROBABILITY MEASURES

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$ , we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$ . Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$ -filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ , which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

Locally n-Connected Compacta and UVn-Maps

We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = {Xα , pβ α , α < β < τ} consisting of compact metrizable LCn-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVn-maps. A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVn) metrizable compacta.

RIGIDITY OF CONTINUOUS QUOTIENTS

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.

Automatic Continuity for Homomorphisms into Free Products

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.

Spaces of measurable functions

For a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Measurements and Gδ-Subsets of Domains

In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain P for which max(P) is a Gδ-subset of P and yet no measurement μ on P has ker(μ) = max(P). We also correct a mistake in the literature asserting that [0, ω1) is a space of this type. We show that if P is a Scott domain and X ⊆ max(P) is a Gδ-subset of P, then X has a Gδ-diagonal and is weakly developable. We show that if X ⊆ max(P) is a Gδ-subset of P, where P is a domain but perhaps not a Scott domain, then X is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P such that max(P) is the usual space of countable ordinals and is a Gδ-subset of P in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.