Topology and its Applications最新文献

In this paper, we give an unstable approach of the May-Lawrence matrix Toda bracket, which becomes a useful tool to detect the unstable phenomena. Then, we give a generalization of the classical isomorphisms between homotopy groups of $(J{S}^m,S^m)$ and $(J{S}^{2m},⁎)$ localized at 2. After that we provide a generalized H-formula for matrix Toda brackets. As an application, we show a new construction of ${\overline{\kappa }}^{\prime }\in {\pi }_{26}\left(S^6\right)$ localized at 2 which improves the construction of ${\overline{\kappa }}^{\prime }$ given by [4].

We prove that the completely irregular set is Baire generic for every non-uniquely ergodic transitive continuous map which satisfies the shadowing property and acts on a compact metric space without isolated points. We also show that, under the previous assumptions, the orbit of every completely irregular point is dense. Afterwards, we analyze the connection between transitivity and the shadowing property, draw a few consequences of their joint action within the family of expansive homeomorphisms, and discuss several examples to test the scope of our results.

Given two discrete Morse functions on a simplicial complex, we introduce the connectedness homomorphism between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in discrete Morse theory at the chain complex level. In particular, we apply it to describe a discrete analogy to ‘cusp-degeneration’ of Morse complexes. A precise comparison between smooth case and our discrete cases is also given.

In this paper, we give a characterization of countably complete topological groups and study when a countably complete subgroup of a topological group is C-embedded. We mainly show that (1) a topological group G is countably complete (the notion introduced by M. Tkachenko in 2012) iff G contains a closed r-pseudocompact subgroup H such that the quotient space $G/H$ is completely metrizable and the canonical quotient mapping $\pi :G\to G/H$ satisfies that ${\pi }^{-1}\left(F\right)$ is r-pseudocompact in G for each r-pseudocompact set F in $G/H$; (2) every countably complete weakly ${\Psi }_{\omega }$-factorizable and ω-balanced subgroup H of a topological group G is C-embedded; (3) every countably complete subgroup H of a pointwise pseudocompact topological group G is C-embedded; (4) every uniformly strongly countably complete and weakly ${\Psi }_{\omega }$-factorizable subgroup H of a topological group G is C-embedded. Further, an ω-narrow locally compact subgroup H of a topological group G is C-embedded.

If X is a hereditarily metacompact ω-scattered space and X has a σ-NSR pair-base at every point of X, then X has a σ-NSR pair-base. If X is a hereditarily meta-Lindelöf ω-scattered space and X has a σ-NSR pair-base at every point of X, then X has property (σ-A). If X is a hereditarily meta-Lindelöf GO-space such that every condensation set of X has property (σ-A), then X has property (σ-A). We point out that there is a gap in the proof of Lemma 37 in [18]. We give a detailed proof for the result. We finally show that if $(X,\tau ,<)$ is a GO-space and $X^{\left(n\right)}$ has property (A) for some $n\in N$, then X has property (A), where $X^{\left(0\right)}=X$, $X^{(i+1)}=\{x\in X^{\left(i\right)}:x$ is not an isolated point of $X^{\left(i\right)}\}$ for each $i<n$. If X is a hereditarily meta-Lindelöf ω-scattered GO-space, then X has a σ-NSR pair-base and $X^{\omega }$ is hereditarily a D-space.

In this paper, some three-space properties of (compactly-fibered, neutrally-fibered) coset spaces are considered. Let $X=G/H$ be a coset space and K a closed subgroup of G with $H\subset K$. It is mainly shown that (1) If $G/H$ is neutrally-fibered, then $G/H$ is second-countable ⇔ $K/H$ and $G/K$ are second-countable; (2) If $G/H$ is compactly-fibered such that all compact (resp., countably compact) subspaces of $K/H$ and $G/K$ are metrizable, then all compact (resp., countably compact) subspaces of $G/H$ are metrizable; (3) If $G/H$ is neutrally-fibered such that $K/H$ is second-countable and $G/K$ an ${\aleph }_0$-space (resp., cosmic), then $G/H$ is an ${\aleph }_0$-space (resp., cosmic); (4) If $G/H$ is neutrally-fibered such that $K/H$ is second-countable and $G/K$ has a star-countable cs-network, then $G/H$ has a star-countable cs-network; (5) If $G/H$ is compactly-fibered such that $K/H$ is locally compact metrizable and $G/K$ stratifiable (resp., k-semi-stratifiable), then $G/H$ is stratifiable (resp., k-semi-stratifiable).

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the ϵ-surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes.

We observe that every zero-dimensional simplicial cochain defines a canonical filtration of a finite simplicial complex and deduce upper estimates for the expected Betti numbers of codimension one random subcomplexes in its support. Moreover, a monotony theorem improves these estimates given any packing of disjoint simplices.

A class of topological spaces is projective (resp., ω-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, (ω-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even ω-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.

We show that it is consistent to have regular closed non-clopen copies of $N^{⁎}$ within $N^{⁎}$ and a non-trivial self-map of $N^{⁎}$ even if all autohomeomorphisms of $N^{⁎}$ are trivial.