We characterize the quasi-metric spaces which have a quasi-metric half-completion and deduce that each paracompact co-stable quasi-metric space having a quasi-metric half-completion is metrizable. We also characterize the quasi-metric spaces whose bicompletion is quasi-metric and it is shown that the bicompletion of each quasi-metric compatible with a quasi-metrizable space is quasi-metric if and only if is finite.
We show that is not normal, if is a limit point of some countable subset of , consisting of points of character . Moreover, such a point is a Kunen point and a super Kunen point.
The Cantor set and the set of irrational numbers are examples of 0-dimensional, totally disconnected, homogeneous spaces which admit elegant characterizations and which play a crucial role in analysis and dynamical systems. In this paper we will start the study of 1-dimensional, totally disconnected, homogeneous spaces. We will provide a characterization of such spaces and use it to show that many examples of such spaces which exist in the literature in various fields are all homeomorphic. In particular,...
For a functor on the category of metrizable compacta, we introduce a conception of a linear functorial operator extending (for each ) pseudometrics from onto (briefly LFOEP for ). The main result states that the functor of -symmetric power admits a LFOEP if and only if the action of on has a one-point orbit. Since both the hyperspace functor and the probability measure functor contain as a subfunctor, this implies that both and do not admit LFOEP.
For some pairs (X,A), where X is a metrizable topological space and A its closed subset, continuous, linear (i.e., additive and positive-homogeneous) operators extending metrics for A to metrics for X are constructed. They are defined by explicit analytic formulas, and also regarded as functors between certain categories. An essential role is played by "squeezed cones" related to the classical cone construction. The main result: if A is a nondegenerate absolute neighborhood retract for metric spaces,...
Some results concerning spaces with countably weakly uniform bases are generalized for spaces with -in-countable ones.
Let a space be Tychonoff product of -many Tychonoff nonsingle point spaces . Let Suslin number of be strictly less than the cofinality of . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification . In particular, this is true if is either or and a cardinal is infinite and not countably cofinal.
We show that given infinite sets and a function which is onto and -to-one for some , the preimage of any ultrafilter of under extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms and a finite-to-one onto function such that for each free ultrafilter of its preimage under does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...