We study closed subspaces of $\kappa$-Ohio complete spaces and, for $\kappa$ uncountable cardinal, we prove a characterization for them. We then investigate the behaviour of products of $\kappa$-Ohio complete spaces. We prove that, if the cardinal ${\kappa }^{+}$ is endowed with either the order or the discrete topology, the space ${\left({\kappa }^{+}\right)}^{{\kappa }^{+}}$ is not $\kappa$-Ohio complete. As a consequence, we show that, if $\kappa$ is less than the first weakly inaccessible cardinal, then neither the space ${\omega }^{{\kappa }^{+}}$, nor the space ${ℝ}^{{\kappa }^{+}}$ is $\kappa$-Ohio complete.

We consider the question of when $X_M=X$, where $X_M$ is the elementary submodel topology on X ∩ M, especially in the case when $X_M$ is compact.

We construct in ZFC an ultrafilter $U\in {\mathbb{N}}^{*}$ such that for every one-to-one function $f:\mathbb{N}\to \mathbb{N}$ there exists $U\in U$ with $f\left[U\right]$ in the summable ideal, i.e. the sum of reciprocals of its elements converges. This strengthens Gryzlov’s result concerning the existence of $0$-points.

A new topological cardinal invariant is defined; it may be considered as a weaker form of the Lindelöf degree.

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

We define a metric $d_S$, called the shape metric, on the hyperspace $2^X$ of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace $(2^{ℝ}2$, dS)$isseparable.Ontheotherhand,wegiveanexampleshowingthat$2ℝ2$isnotseparableinthefundamentalmetricintroducedbyBorsuk.$

In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions,...

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a naturally occuring example of a quotient map such that q × q fails to be a quotient map. With the quotient topology, this example shows π₁(X,p) can fail to be a topological group if X is locally path connected.