This probably doesn’t deserve a § all to itself, but it will be handy to know it in what follows. Recall that a set S in a metric (or topological) space X is dense-in-itself 1 if every neighborhood of each s0 ∈ S contains points s ∈ S different from s0. One says that S is perfect if it is closed and dense-in-itself. A topological (or metric) space is separable if it has a countable subset whose...

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Ru...

We show that many expanding circle maps have a shrinking target property, but that circle homeomorphisms and isometries of complete, separable metric spaces do not.

In this note we show, following W. Holsztynski, that there is a continuous metric d on R such that any finite metric space is isometrically embeddable into (R, d). Let M be a family of metric spaces. A metric space U is said to be a universal space for M if any space from M is (isometrically) embeddable in U . Fréchet [2] proved that l (the space of all bounded sequences of real numbers endowed...

A space X is cometrizable if X has a coarser metric topology such that each point of X has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space X : (a) X ...

In this paper, we show that a regular space with a locally countable weak-base is g-metrizable. Secondly, we establish the relationships between spaces with a locally countable weak-base (resp. spaces with a locally countable weak-base consisting of א0-subspaces) and metric spaces (resp. locally separable metric spaces) by means of compact-covering maps, 1-sequence-covering maps, compact maps, ...

We show that every complete metric space is homeomorphic to the locus of zeros of an entire analytic map from a complex Hilbert space to a complex Banach space. As a corollary, every separable complete metric space is homeomorphic to the locus of zeros of an entire analytic map between two complex Hilbert spaces. §1. Douady had observed [8] that every compact metric space is homeomorphic to the...

It is consistent with the axioms of set theory that for every metric space X which is isometric to some separable Banach space or to Urysohn’s universal separable metric space U the following holds: (?)X There exists a nowhere meager subspace of X of cardinality א1 and any two nowhere meager subsets of X of cardinality א1 are almost isometric to each other. As a corollary, it is consistent that...

In this paper, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces. Extensions to general symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper. AMS 2000 Mathematics Subject Classification: Primary 60J45, 3...

We shall discuss further regularity properties of null-additive fuzzy measure on metric spaces following the previous results. Under the null-additivity condition, some properties of the inner/outer regularity and the regularity of fuzzy measure are shown. Also the strong regularity of fuzzy measure is discussed on complete separable metric spaces. As an application of strong regularity, we pre...