Topology and its Applications最新文献

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A computational framework for weighted simplicial homology

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109177

Andrei C. Bura , Neelav S. Dutta , Thomas J.X. Li , Christian M. Reidys

We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring

R = F [[π]]

. This is achieved by starting from a basis for classical homology of the n-th skeleton for the underlying complex with coefficients in the residue field

F

and then lifting it to a basis for the weighted homology with coefficients in the ring R. Using the latter, a bijection is established between

n + 1

and n dimensional simplices whose weight ratios provide the exponents of the π-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over R. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of R, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently.

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引用次数: 0

Metrizable spaces homeomorphic to the hyperspace of nonblockers of singletons of a continuum

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-02-01 DOI: 10.1016/j.topol.2024.109151

David Maya, Fernando Orozco-Zitli, Emiliano Rodríguez-Anaya

A continuum is a nondegenerate compact connected metric space. The hyperspace of all nonempty closed subsets of a continuum X topologized by the Hausdorff metric is denoted by

2^{X}

. Given a continuum X, the subspace

NB (F_{1} (X))

2^{X}

consists of all elements

A \in 2^{X} - {X}

such that for each

x \in X - A

, the union of all subcontinua of X containing x and contained in

X - A

is a dense subset of X. The members of

NB (F_{1} (X))

are called nonblocker subsets of the singletons of the continuum X. In this paper, we show that each proper nonempty open subset U of a compact metric space can be embedded in a continuum X such that U and the hyperspace of nonblocker subsets of X are homeomorphic. This answers a question posed by J. Camargo, F. Capulín, E. Castañeda-Alvarado and D. Maya.

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引用次数: 0

Fréchet spaces, ω-Rudin property and Smyth power spaces

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.topol.2025.109235

Xiaoquan Xu , Hualin Miao , Qingguo Li

For a

T_{0}

-space X, let

K (X)

be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order (i.e., the reverse inclusion order). The Smyth power poset

K (X)

equipped with the upper Vietoris topology is called the Smyth power space of X and is denoted by

P_{S} (X)

. This paper is mainly devoted to discuss some properties of Fréchet spaces from the viewpoint of intersection of topology and domain theory. We prove that if the sobrification (especially, the Hoare power space) of a

T_{0}

-space X is a Fréchet space, then X is an ω-Rudin space. Hence every ω-well-filtered space for which its sobrification (especially, its Hoare power space) is a Fréchet space is sober, and every second-countable ω-well-filtered space is sober. We also show that if the Smyth power space of a

T_{0}

-space X is a Fréchet space, then the Scott topology is coarser than the upper Vietoris topology on

K (X)

, whence if X is additionally well-filtered, then the topology of Smyth power space

P_{S} (X)

is the Scott topology of

K (X)

. Moreover, if X is a second-countable ω-well-filtered space, then the topology of

P_{S} (X)

is the Scott topology of

K (X)

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引用次数: 0

Every Lindelöf scattered subspace of a Σ-product of real lines is σ-compact

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.topol.2025.109234

Antonio Avilés , Mikołaj Krupski

We prove that every Lindelöf scattered subspace of a Σ-product of first-countable spaces is σ-compact. In particular, we obtain the result stated in the title. This answers some questions of Tkachuk (2022) [9].

引用次数: 0

The L-algebras related to prime spectra of Bézout domains and abelian ℓ-groups

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-28 DOI: 10.1016/j.topol.2025.109231

Wolfgang Rump

The ℓ-spectrum problem asks for a topological characterization of the prime spectrum of a Bézout domain (equivalently, the inverse prime spectrum of an abelian ℓ-group). While a general solution is out of reach, the analogous problem for the maximal spectrum of a Bézout domain was solved in a previous article. An analysis of the ℓ-spectrum problem by means of L-algebras is given. If the prime spectrum is an Esakia space, the known explicit solutions will be compared and related to a finitely additive measure that connects two fundamental classes of L-algebras. The abelian ℓ-groups constructed by several authors from an Esakia space are shown to be structure groups of L-algebras. The L-algebraic method is then extended to more general prime spectra, which leads to a new sufficient criterion for spectral spaces to be representable as prime spectra of Bézout domains.

引用次数: 0

Identities for Whitehead products and infinite sums

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.topol.2025.109232

Jeremy Brazas

Whitehead products and natural infinite sums are prominent in the higher homotopy groups of the n-dimensional infinite earring space

E_{n}

and other locally complicated Peano continua. In this paper, we derive general identities for how these operations interact with each other. As an application, we consider a shrinking wedge

of finite

(n - 1)

-connected CW-complexes and compute the infinite-sum closure

W_{2 n - 1} (X)

of the set of Whitehead products

[α, β]

π_{2 n - 1} (X)

where

α, β \in π_{n} (X)

are represented in respective sub-wedges that meet only at the basepoint. In particular, we show that

W_{2 n - 1} (X)

is canonically isomorphic to

\prod_{j = 1}^{\infty} (π_{n} (X_{j}) \otimes \prod_{k > j} π_{n} (X_{k}))

. The insight provided by this computation motivates a conjecture about the isomorphism type of the elusive groups

π_{2 n - 1} (E_{n})

n \geq 2

引用次数: 0

From λ-hollow frames to λ-repletions in W: II. λ-repletions in W

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.topol.2025.109233

Richard N. Ball , Anthony W. Hager , Joanne Walters-Wayland

<div><div>In this article we analyze the fine structure of the essential extensions of an object of W, the category of divisible archimedean lattice ordered groups with designated weak units. In particular, we show that an object G has an ordinally indexed sequence <math><msub><mrow><mo>{</mo><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>}</mo></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></msub></math> of essential extensions with the following features.<img><ul><li>•<div><math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math> is (isomorphic to) the identity function on G.</div></li><li>•<div>For every <math><mi>α</mi><mo>></mo><mn>0</mn></math>, <math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math> is an essential extension of G into a W-object which is of the form <math><mi>R</mi><mi>L</mi></math> for some frame L, and which is λ-replete for some λ.</div></li><li>•<div>Every such extension is (isomorphic to) <math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math> for a unique α.</div></li><li>•<div><math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><msub><mrow><mi>δ</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></msubsup></math> is (isomorphic to) the maximal essential extension of G.</div></li><li>•<div>If <math><mi>λ</mi><mo>≤</mo><mi>ν</mi><mo>≤</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>G</mi></mrow></msub></math> then <math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>ν</mi></mrow></msubsup></math> factors through <math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>λ</mi></mrow></msubsup></math>.</div></li></ul> Here a W-object is said to be λ-replete if it has the following equivalent properties.<ul><li>•<div>Every λ-generated W-kernel is a polar.</div></li><li>•<div>Every proper λ-generated W-kernel of G is contained in a proper polar.</div></li><li>•<div>For λ-generated W-kernels <math><msub><mrow><mi>K</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, if <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊈</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math> then there exists <math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></m

引用次数: 0

On strong generalized topology generated by the porosity

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-23 DOI: 10.1016/j.topol.2025.109223

Jacek Hejduk, Piotr Nowakowski

This paper presents the strong generalized topology of sets having the property that every element is the porosity point of its complement. In the frame of generalized topology the fundamental topological properties are investigated including the separation axioms. There is the aspect of delving into the study of continuity including the notion of the approximate continuity.

引用次数: 0

Countably tight, zero-dimensional and totally disconnected dense subspaces of Cp(X)

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.topol.2025.109220

Joel Aguilar-Velázquez , Reynaldo Rojas-Hernández

In this paper we prove that under CH there exists a space X such that

C_{p} (X)

does not admit dense subspaces of countable tightness, partially answering Problem 1 of [13]. We prove that if X has cardinality at most continuum, then

C_{p} (X)

contains a dense zero-dimensional subspace and a dense totally disconnected non zero-dimensional subspace. We also provide an example of a compact space X such that

C_{p} (X, 2)

is exponentially separable but X is not Corson compact.

引用次数: 0

Brouwer's fixed-point theorem on the finite product of compact linearly ordered topological spaces

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-01-21 DOI: 10.1016/j.topol.2025.109221

Tetsuya Ishiu

We shall show that the finite product of compact connected linearly ordered topological spaces satisfies Brouwer's fixed-point theorem and the Poincaré-Miranda Theorem.

引用次数: 0

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