Topology and its Applications最新文献

We revisit the application of Shelah's Revised GCH Theorem [19] to diamond. We also formulate a generalization of the theorem and prove a small fragment of it. Finally we consider another application of the theorem, to covering numbers of the form $\mathrm{cov}(-,-,-,\omega )$.

Menasco proved that if G is a reduced, alternating, connected diagram of a link L and G is prime then L is prime. This surprising and important result has been generalized to other classes of links, as well as to tangles and spatial graphs. After exploring some issues with previous results, we obtain new splitting results for tangles and spatial graphs.

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y$, denoted $T{C}_r\left(f\right)$, that interacts with $T{C}_r\left(X\right)$ and $T{C}_r\left(Y\right)$ in the same way Jamie Scott's topological complexity map $TC\left(f\right)$ interacts with $TC\left(X\right)$ and $TC\left(Y\right)$. Furthermore, we apply $T{C}_r\left(f\right)$ to studying group homomorphisms $\varphi :\Gamma \to \Lambda$.

In addition, we give the characterization of cohomological dimension of group homomorphisms.

In this paper, we develop the new method to compute the homotopy groups of the mapping cone $C_f=Y{\cup }_{f}CX$ beyond the metastable range by analysing the homotopy of the n-th filtration of the relative James construction $J(X,A)$ for CW-pair $A\stackrel{i}{\hookrightarrow }X$, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map $X{\cup }_{i}CA\to \Sigma A$. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod $2^r$ Moore spaces for all positive integers r.

This paper includes two main results. Dual discreteness is a well known generalization of D-spaces. The first one is that every Σ-product of compact metric spaces is dually discrete. The property aD is another generalization of D-spaces, and it implies irreducibility. The second one is that the product $N^{{\omega }_1}$ of ${\omega }_1$ many copies of $N$ is irreducible, where $N$ denotes an infinite countable discrete space.

We prove that:

• 1.

  If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable.

• 2.

  A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally Hausdorff and has a countable pseudocharacter.

• 3.

  Let Y be a dense-in-itself space with the following property: $\forall y\in Y\exists Q\left(y\right)\subseteq Y\left[y\text{ is a non-isolated q-point in }Q\right(y\left)\right]$. If Y is an open-closed image of a submetrizable space, then Y is submetrizable.

• 4.

  There exist a submetrizable space X, a regular hereditarily paracompact non submetrizable first-countable space Y, and an open-closed map $f:X\to Y$.

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].