In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${[0,1]}^{\infty }$ and a Cantor set, respectively.

In this note we study the relation between $k_R$-spaces and $k$-spaces and prove that a $k_R$-space with a $\sigma$-hereditarily closure-preserving $k$-network consisting of compact subsets is a $k$-space, and that a $k_R$-space with a point-countable $k$-network consisting of compact subsets need not be a $k$-space.

For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc\left(X\right)\to Pc\left(FX\right)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $S{P}_G^n$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\cdots ,n\}$ has a one-point orbit. Since both the hyperspace functor $exp$ and the probability measure functor $P$ contain $S{P}^2$ as a subfunctor, this implies that both $exp$ and $P$ 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,...

We call a function $f:X\to Y$ P-preserving if, for every subspace $A\subset X$ with property P, its image $f\left(A\right)$ also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, $T_1$ range, and is connectedness-preserving...

For a non-isolated point $x$ of a topological space $X$ let ${\mathrm{nw}}_{\chi }\left(x\right)$ be the smallest cardinality of a family $𝒩$ of infinite subsets of $X$ such that each neighborhood $O\left(x\right)\subset X$ of $x$ contains a set $N\in 𝒩$. We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$; (b) for each point $x\in X$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$ there is an injective sequence ${\left(x_n\right)}_{n\in \omega }$ in $X$ that $ℱ$-converges to $x$ for some meager filter $ℱ$ on $\omega$; (c) if a functionally Hausdorff space $X$ contains an $ℱ$-convergent injective sequence for some meager filter...

In this article, we extend the work on minimal Hausdorff functions initiated by Cammaroto, Fedorchuk and Porter in a 1998 paper. Also, minimal Urysohn functions are introduced and developed. The properties of heredity and productivity are examined and developed for both minimal Hausdorff and minimal Urysohn functions.

Let $ℛL$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛL$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $ℛL$ is isomorphic to the $f$-ring $C\left(\Sigma L\right)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is...