The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $\mathscr{H}$ be a Hilbert space and let $A$ be a positive bounded operator on $\mathscr{H}$. The semi-inner product ${\langle h\mid k\rangle }_A:=\langle Ah\mid k\rangle$, $h,k\in \mathscr{H}$, induces a semi-norm ${\parallel ·\parallel }_A$. This makes $\mathscr{H}$ into a semi-Hilbertian space. An operator $T\in {ℬ}_A\left(\mathscr{H}\right)$ is said to be $(n,m)$-$A$-normal if $[T^n,{\left(T^{{♯}_A}\right)}^m]:=T^n{\left(T^{{♯}_A}\right)}^m-{\left(T^{{♯}_A}\right)}^m{T}^n=0$ for some positive integers $n$ and $m$.

In this paper, we give characterizations of certain weak-open images of metric spaces.

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...