If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a metric extension of $X$. If $T_1$ and $T_2$ are metric extensions of $X$ and there is a continuous map of $T_2$ into $T_1$ keeping $X$ pointwise fixed, we write $T_1\le T_2$. If $X$ is noncompact and metrizable, then $\left(ℳ\right(X),\le )$ denotes the set of metric extensions of $X$, where $T_1$ and $T_2$ are identified if $T_1\le T_2$ and $T_2\le T_1$, i.e., if there is a homeomorphism of $T_1$ onto $T_2$ keeping $X$ pointwise fixed. $\left(ℳ\right(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [The structure of...

In this paper we define for fuzzy topological spaces a notion corresponding to proto-metrizable topological spaces. We obtain some properties of these fuzzy topological spaces, particularly we give relations with non-archimedean, and metrizable fuzzy topological spaces.

Soit $X$ un espace topologique régulier et fortement $\alpha$-favorable : si $X$ est image continue d’un espace métrisable séparable alors $X$ est lusinien; ceci répond à une question de R. Haydon. Si $X$ est seulement de Lindelöf et à diagonale $G_{\delta }$ alors l’espace mesurable $(X,Ba(X\left)\right))$ est standard; on en déduit que si l’ensemble des points extrêmaux d’un convexe compact $K$ est de Lindelöf et à diagonale $G_{\delta }$, alors $K$ est métrisable.

Let $(X,\rho )$, $(Y,\sigma )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\sigma (f\left(x\right),f\left(y\right))/\rho (x,y)$; $x,y\in X$, $x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\epsilon >0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\to Z$ ($Z$ arbitrary metric space) with $dist\left(f\right)<K$ one can find a mapping $g:X\to Y$, such that both the mappings $g$ and ${f|}_{g\left(X\right)}$ have distortion at...

We continue the study of remainders of metrizable spaces, expanding and applying results obtained in [Fund. Math. 215 (2011)]. Some new facts are established. In particular, the closure of any countable subset in the remainder of a metrizable space is a Lindelöf p-space. Hence, if a remainder of a metrizable space is separable, then this remainder is a Lindelöf p-space. If the density of a remainder Y of a metrizable space does not exceed $2^{\omega }$, then Y is a Lindelöf Σ-space. We also show that many of...

In this paper, we prove that each sequence-covering and boundary-compact map on $g$-metrizable spaces is 1-sequence-covering. Then, we give some relationships between sequence-covering maps and 1-sequence-covering maps or weak-open maps, and give an affirmative answer to the problem posed by F.C. Lin and S. Lin in [Lin.F.C.and.Lin.S-2011].

Guoliang Yu has introduced a property on discrete metric spaces and groups, which is a weak form of amenability and which has important applications to the Novikov conjecture and the coarse Baum–Connes conjecture. The aim of the present paper is to prove that property in particular examples, like spaces with subexponential growth, amalgamated free products of discrete groups having property A and HNN extensions of discrete groups having property A.

In this paper semigroups of contractions of metric spaces are considered. The semigroup of contractions of the wreath product of metric spaces is calculated.