arXiv subjects
Publications and source records attributed to Vlad Smolin.
We investigate a question posed by Huaipeng Chen: if $X$ and $Y$ are paracompact submetrizable spaces and $f:X\to Y$ is a perfect map, can $X$ and $Y$ be submetrized by metrics $\rho$ and $d$ respectively such that $f$ remains perfect with respect to the induced topologies?
Answering a question posed by Vladimir Tkachuk, we prove that every $T_1$ connected first-countable space is a continuous open image of a connected metrizable space.
In [5] we studied spaces with a Lusin $\pi$-base and $\pi$-spaces and posed the following question: Does the class of continuous open images of spaces with a Lusin $\pi$-base equal the class of continuous open images of $\pi$-spaces? We give a negative answer to this question.
We prove that: 1. If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable. 2. A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally Hausdorff and has a countable pseudocharacter. 3. Let $Y$ be a dense-in-itself space with the following property: $\forall y\in Y\ \exists Q(y) \subseteq Y\ [y \text{ is a non-isolated q-point in } Q(y)]$. If $Y$ is an open-closed image of a submetrizable space, then $Y$ is submetrizable. 4. There exist a submetrizable space $X$, a regular hereditarily paracompact non submetrizable first-countable space $Y$, and an open-closed map $f\colon X \to Y$.
A topological space is called a submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.
We study spaces that can be mapped onto the Baire space (i.e. the countable power of the countable discrete space) by a continuous quasi-open bijection. We give a characterization of such spaces in terms of Souslin schemes and call these spaces $\pi$-spaces. We show that every space that has a Lusin $\pi$-base is a $\pi$-space and that every second-countable $\pi$-space has a Lusin $\pi$-base. The main result of this paper is a characterization of continuous open images of $\pi$-space.
In this note we prove that a regular continuous open image of the Sorgenfrey line with an uncountable weight has a closed subspace that is homeomorphic to the Sorgenfrey line. As a corollary we deduce the theorem in the title.
We give a description of Hausdorff continuous open images of the Sorgenfrey line: these are precisely those spaces that have a Sorgenfrey base. Using this description we prove that no Hausdorff compact space that contains a copy of the Sorgenfrey line is a continuous open image of it; in particular the double-arrow space is not a continuous open image of the Sorgenfrey line.