In this paper $ss$-quotient maps and $ssq$-spaces are introduced. It is shown that (1) countable tightness is characterized by $ss$-quotient maps and quotient maps; (2) a space has countable tightness if and only if it is a countably bi-quotient image of a locally countable space, which gives an answer for a question posed by F. Siwiec in 1975; (3) $ssq$-spaces are characterized as the $ss$-quotient images of metric spaces; (4) assuming $2^{\omega }<2^{{\omega }_1}$, a compact $T_2$-space is an $ssq$-space if and only if every countably compact subset...

We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber $f^{-1}\left(y\right)$ belongs to a class S of spaces, then there exists an $F_{\sigma }$-set A ⊂ X such that A ∈ S and $dim{f}^{-1}\left(y\right)\setminus A=0$ for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all $g\in C\left(X{,}^{n+1}\right)$.

We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally ${\aleph }_1$-presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-${\aleph }_0$-generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include...

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.