Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.07744

Homotopy properties of the space $OS_{f}(X)$

Kh. ~Kh. ~Kurbanov·A. ~Ya. ~Ishmetov
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a given compact Hausdorff space $X$, we construct the space $OS_{f}(X)$ of normed, order-preserving, weakly additive, positively homogeneous and semi-additive functionals (for brevity, semi-additive functionals) and it is proved that the hyperspace $\exp\, X$ of the space $X$ is a deformation retract of the constructed space. Further we show that the shapes of the spaces $OS_{f}(X)$ and $\exp\, X$ coincide. We establish that if the space $\exp\, X$ is contractible, then the space $OS_{f}(X)$ is also contractible.

Explore related subjects

math.GNmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kh. ~Kh. ~Kurbanov, A. ~Ya. ~Ishmetov. 2020-03-17. Homotopy properties of the space $OS_{f}(X)$. https://arxiv.org/abs/2003.07744

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

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous linearly ordered spaces

Every compact subset of a homogeneous generalized ordered (GO) space has character at most $ω_1$ and cardinality at most $2^{ω_1}$; if such a subset has uncountable character, then the character of the whole space equals $ω_1$ and its $π$-character is countable. We construct a homogeneous $σ$-compact linearly ordered space (LOTS) $\mathbf{H}$ containing a compact subset $\mathbf{S}$ of cardinality $2^{ω_1}$ whose character is $ω_1$ at every point and whose weight and Souslin number are both $2^{ω_1}$; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a $P$-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a $P$-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.

math.GN↗
arXiv2026

A New Approach to Universal Measurability

V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_τ(X)$ is $C$-embedded in $P_σ(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(βX)$ if and only if $X$ is pseudocompact, where $P_σ$, $P_τ$, and $P_R$ are the functors of probability $σ$-additive on the Baire $σ$-algebra, $τ$-additive, and Radon measures on the space $X$? The answers to these questions are negative. However, if instead of probability measures we consider the corresponding alternating measures $M_σ$, $M_τ$, and $M_R$, the situation changes. It is proved that (i) $M_τ(X)$ is $C$-embedded in $M_σ(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(βX)$ if and only if $X$ is pseudocompact. The question of $C$-embedding of measure spaces is an extension of the question of coincidence of measure spaces, which is a development of the classical concepts of universally measurable and universal measure zero sets. A general theorem is obtained, which leads to the mentioned results.

math.GN↗
arXiv2026

One-Point Metrizable Coarsenings: Gauges and Local Metric Preservation

Let $(X,τ)$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $σ\subseteqτ$ that agree with $τ$ on $X\setminus\{a\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,τ)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ can be made to satisfy $p(a,x_n)\leλ_n$ for any positive null sequence $(λ_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.

math.GN↗