According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_{\delta }$-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its...

For a Tychonoff space $X$, let $\downarrow {\mathrm{C}}_F\left(X\right)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable but $X$ is not first-countable.

A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.

The aim of the paper is to introduce and study a new class of spaces called mildly ${(1,2)}^{*}$-normal spaces and a new class of functions called ${(1,2)}^{*}$-$\mathrm{rg}$-continuous, ${(1,2)}^{*}$-$\mathrm{R}$-map, almost ${(1,2)}^{*}$-continuous function and almost ${(1,2)}^{*}$-$\mathrm{rg}$-closed function in bitopological spaces. Subsequently, the relationships between mildly ${(1,2)}^{*}$-normal spaces and the new bitopological functions are investigated. Moreover, we obtain characterizations of mildly ${(1,2)}^{*}$-normal spaces, properties of the new bitopological functions and preservation theorems for...

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $\Phi \subset 2^{2^X}$, the space of maximal chains in $2^X$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo\left(X\right)}$, the universal minimal space of Homeo(X), is not transitive (improving...

We prove that an ultrametric space can be bi-Lipschitz embedded in ${ℝ}^n$ if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.

Let f: (X,A) → (X,A) be a relative map of a pair of compact polyhedra. We introduce a new relative homotopy invariant $N^C(f;X,A)$, which is a lower bound for the component numbers of fixed point sets of the self-maps in the relative homotopy class of f. Some properties of $N^C(f;X,A)$ are given, which are very similar to those of the relative Nielsen number N(f;X,A).