We prove that each analytic set in ℝⁿ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.

We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following: ...

2000 Mathematics Subject Classification: 06A06, 54E15An ordered pair X(R) = ( X, R ) consisting of a nonvoid set X and a nonvoid family R of binary relations on X is called a relator space. Relator spaces are straightforward generalizations not only of uniform spaces, but also of ordered sets. Therefore, in a relator space we can naturally define not only some topological notions, but also some order theoretic ones. It turns out that these two, apparently quite different, types of notions are closely...

The aim of the paper is to prove that the bounded and unbounded Urysohn universal spaces have unique (up to isometric isomorphism) structures of metric groups of exponent 2. An algebraic-geometric characterization of Boolean Urysohn spaces (i.e. metric groups of exponent 2 which are metrically Urysohn spaces) is given.

We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $𝔸=\langle A,F,E\rangle$ (it means that $\langle A,F\rangle$ is a semigroup and $E$ is an automorphism or an antiautomorphism on $\langle A,F\rangle$ such that $E\circ E=\mathrm{𝐈𝐝}\upharpoonright A$) is, generally, a sequence $𝔸\upharpoonright B$, $𝔸\upharpoonright U_c$, $c\in \mathrm{𝐅𝐙}$ (where $\mathrm{𝐅𝐙}$ is the class of finite integers) of substructures of $𝔸$ such that $B\subseteq U_c\subseteq U_d$ holds for each $c\le d$. We denote this line as $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ and we say that a mapping $H$ is a valuation of the line $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ in a line $\widehat{𝔸}{({\widehat{U}}_c,\widehat{B})}_{c\in \mathrm{𝐅𝐙}}$ if it is, for each $c\in \mathrm{𝐅𝐙}$, a valuation of the triad $𝔸(U_c,B)$ in $\widehat{𝔸}({\widehat{U}}_c,\widehat{B})$. Some theorems on an existence of...

Various characterizations of realcompactness are transferred to uniform spaces giving non-equivalent concepts. Their properties, relations and characterizations are described in this paper. A Shirota-like characterization of certain uniform realcompactness proved by Garrido and Meroño for metrizable spaces is generalized to uniform spaces. The paper may be considered as a unifying survey of known results with some new results added.

2000 Mathematics Subject Classification: 54H25, 47H10.The concept of semi compatibility is given in probabilistic metric space and it has been applied to prove the existence of unique common fixed point of four self-maps with weak compatibility satisfying an implicit relation. At the end we provide examples in support of the result.Authors thank to MPCOST, Bhopal for financial support through the project M-19/2006.

2000 Mathematics Subject Classification: 47H10, 54E15.The purpose of this paper is to define the notion of A-distance and E-distance in uniform spaces and give several new common fixed point results for weakly compatible contractive or expansive selfmappings of uniform spaces.

We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if $(G,·,\tau )$ is a regular right (left) semitopological group with $\mathrm{dev}\left(G\right)<\mathrm{Nov}\left(G\right)$ such that all left (right) translations are feebly continuous, then $(G,·,\tau )$ is a topological group. This extends several results in literature.