On some classes of Lindelöf Sigma-spaces

Wieslaw Kubis; Oleg Okunev; Paul J. Szeptycki

doi:10.1016/j.topol.2005.09.009

Home/Archive/arXiv

arXiv · math/0511606

On some classes of Lindelöf Sigma-spaces

Wieslaw Kubis·Oleg Okunev·Paul J. Szeptycki

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider special subclasses of the class of Lindelöf Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $LΣ(\leqκ)$ if it admits a cover by compact subspaces of weight $κ$ and a countable network for the cover. We restrict our attention to $κ\leqω$. In the case $κ=ω$, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the $P$-approximable spaces considered by Tkacenko. The case $κ=1$ corresponds to the spaces of countable network weight, but even the case $κ=2$ gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight $\aleph_1$ is in the class $LΣ(\leq ω)$, answering a question of Tkachuk. As well, we study whether certain compact spaces in $LΣ(\leqω)$ have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.

Explore related subjects

math.GN

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Wieslaw Kubis, Oleg Okunev, Paul J. Szeptycki. 2005-11-24. On some classes of Lindelöf Sigma-spaces. https://doi.org/10.1016/j.topol.2005.09.009

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The circle as a topological fractal

We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.

math.GN↗

arXiv2026

Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds

Let $(X,τ)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $σ\subseteqτ$ agreeing with $τ$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,τ)$ is compact, in which case it consists only of $τ$.

math.GN↗

arXiv2026

Journey into special $T_1$-spaces

In this survey, we review certain types of special $T_{1}$ spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary $T_{0}$ spaces. Furthermore, the $T_{1}$ separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.

math.GN↗