Let $\left\{{𝒫}_n\right\}$ be a sequence of covers of a space $X$ such that $\left\{st(x,{𝒫}_n)\right\}$ is a network at $x$ in $X$ for each $x\in X$. For each $n\in \mathbb{N}$, let ${𝒫}_n=\{P_{\beta }:\beta \in {\Lambda }_n\}$ and ${\Lambda }_n$ be endowed the discrete topology. Put $M=\{b=\left({\beta }_n\right)\in {\Pi }_{n\in \mathbb{N}}{\Lambda }_n:\left\{P_{{\beta }_n}\right\}$ forms a network at some point $x_{b}inX\}$ and $f:M⟶X$ by choosing $f\left(b\right)=x_b$ for each $b\in M$. In this paper, we prove that $f$ is a sequentially-quotient (resp. sequence-covering, compact-covering) mapping if and only if each ${𝒫}_n$ is a $c{s}^{*}$-cover (resp. $fcs$-cover, $cfp$-cover) of $X$. As a consequence of this result, we prove that $f$ is a sequentially-quotient, $s$-mapping if and only if it is...

Trivially symmetrizable, trivially semi-metrizable and trivially D-completely regular mappings are defined. They are characterized as mappings parallel to symmetrizable, semi-metrizable and D-completely regular spaces correspondently. One shows that trivially D-completely regular mappings, i.e. submappings of fibrewise products of developable mappings coincide (up to homeomorphisms) with submappings of fibrewise products of semi-metrizable mappings.

In this paper, we give some characterizations of metric spaces under weak-open $\pi$-mappings, which prove that a space is $g$-developable (or Cauchy) if and only if it is a weak-open $\pi$-image of a metric space.

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. The behavior of $\pi$-metrizable spaces under certain types of mappings is studied. In particular we characterize strongly $d$-separable spaces as those which are the image of a $\pi$-metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a $\pi$-metrizable space under an open continuous mapping. A question posed by Arhangel’skii regarding if a $\pi$-metrizable topological group must be metrizable receives...

Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We...

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].