Topology and its Applications最新文献

In this work, we are concerned with the realization of spaces up to rational homotopy as classifying spaces. In this paper, we first show that a class of rank-two rational spaces cannot be realized up to rational homotopy as the classifying space of any n-connected and π-finite space for $n\ge 1$. We also show that the Eilenberg-Mac Lane space $K(Q^r,n)$ $(r\ge 2,n\ge 2)$ can be realized up to rational homotopy as the classifying space of a simply connected and elliptic space X if and only if X has the rational homotopy type of ${\prod }_r{S}^{n-1}$ with n even.

The star versions of the selection principle $U_{\mathrm{fin}}(O,\Omega )$, namely $U_{\mathrm{fin}}^{⁎}(O,\Omega )$, $S{S}_{\mathrm{fin}}^{⁎}(O,\Omega )$ and ${}^{⁎}U_{\mathrm{fin}}(O,\Omega )$ are studied. We explore ramifications concerning critical cardinalities. Quite a few interesting observations are obtained while dealing with the Isbell-Mrówka spaces, Niemytzki plane and Alexandroff duplicates. Properties like monotonically normal and locally countable cellularity (introduced here) play an important role in our investigation. We study games corresponding to $U_{\mathrm{fin}}(O,\Omega )$ and its star variants which have not been investigated in prior works. Some open problems are posed.

In this paper, we introduce the notion of weakly ω-balanced semitopological groups and prove that the class of weakly ω-balanced semitopological groups is closed under taking subgroups and products. It is prove that a regular (Hausdorff, $T_1$) semitopological group G admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, $T_1$) semitopological groups with a weak development if and only if G is weakly ω-balanced and $Ir\left(G\right)\le \omega$ ($Hs\left(G\right)\le \omega$, $Sm\left(G\right)\le \omega$).

We give a new combinatorial description of the cohomology ring structure of $H^{⁎}\left(M\right(A);Z)$ of the complement $M\left(A\right)$ of a real complexified toric arrangement $A$ in ${\left(C^{⁎}\right)}^d$. In particular, we correct an error in the paper [4].

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).