相关论文: Countable Toronto spaces

A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable…

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example…

We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism.

A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty…

For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we…

A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and…

A topological space $X$ is a $\Delta$-space (or $X \in \Delta$) if for any decreasing sequence $\{A_n : n < \omega\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < \omega\}$ of open sets with empty…

We prove the following results. 1. If $X$ is a $\alpha$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f:X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

We use a natural forcing to construct a left-separated topology on an arbitrary cardinal kappa. The resulting left-separated space X_kappa is also 0-dimensional T_2, hereditarily Lindelof, and countably tight. Moreover if kappa is regular…

A topological space $X$ is $\mathbb R^{\omega_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{\omega_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

We prove that a $T_0$ topological space is $\omega$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this,…

What topological spaces can be partitioned into copies of the Cantor space $2^\omega$? An obvious necessary condition is that a space can be partitioned into copies of $2^\omega$ only if it can be covered with copies of $2^\omega$. We prove…

Metrizable spaces are studied in which every closed set is an $\alpha$-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of…

A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-isolated point are reversible. By Stone duality, these spaces…

The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mathcal{L}_{\omega_1 \omega}$)…

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if kappa^{<kappa}=kappa then there is such a space of height kappa^+ with…

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^\kappa$ the image $f(X)$ is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying…

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological…