Topology and its Applications最新文献

We give a family of slice-torus invariants ${\widetilde{ss}}_c$, each defined from the c-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements c in any principal ideal domain R. For the special case $(R,c)=\left(F\right[H],H)$ where F is any field, we prove that ${\widetilde{ss}}_c$ coincides with the Rasmussen invariant $s^F$ over F. Compared with the unreduced invariants $s{s}_c$ defined by the first author in a previous paper, we prove that $s{s}_c={\widetilde{ss}}_c$ for $(R,c)=\left(F\right[H],H)$ and $(Z,2)$. However for $(R,c)=(Z,3)$, computational results show that $s{s}_3$ is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.

We prove that for any nontrivial knot $K\subset S^3$ and a $p/q$-cable knot $K^{\star }$ of K, the tunnel number $t\left(K\right)=t\left(K^{\star }\right)$ if and only if K is $p/q$-primitive. This result solves a problem mentioned in [8].

This article studies different topological properties of the space of maximal d-elements of an M-frame with a unit. We characterize when the space $Max\left(dL\right)$ is Hausdorff, answering the question posed in [2]. We also characterize other topological properties of $Max\left(dL\right)$, namely zero-dimensional, discrete, and clopen π-base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of d-semiprime frames.

Following the theory of tensor triangular support introduced by Sanders, which generalizes the Balmer-Favi support, we prove the local version of the result of Zou that the Balmer spectrum being Hochster weakly scattered implies the local-to-global principle.

That is, given an object t of a tensor triangulated category $T$ we show that if the tensor triangular support $\text{Supp}\left(t\right)$ is a weakly scattered subset with respect to the inverse topology of the Balmer spectrum $\text{Spc}\left(T^c\right)$, then the local-to-global principle holds for t.

As immediate consequences, we have the analogue adaptations of the well-known statements that the Balmer spectrum being noetherian or Hausdorff scattered implies the local-to-global principle.

We conclude with an application of the last result to the examination of the support of injective superdecomposable modules in the derived category of an absolutely flat ring which is not semi-artinian.

The class of SHD spaces was recently introduced in [12]. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space X such that $F\left[X\right]$, the Pixley-Roy hyperspace of X, βX, the Stone-Čech compactification of X, and $C_p\left(X\right)$, the ring of continuous functions over X equipped with the topology of pointwise convergence, are SHD.

In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between X and $F\left[X\right]$ regarding these new concepts.

The purpose of this note is to start the systematic analysis of cofinal types of topological groups.

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy $h_{QU}\left(\psi \right)$ for a uniformly continuous self-map ψ of a quasi-metric or a quasi-uniform space X. In this paper, we discuss the connection between the topological entropy functions $h,h_f$ and the quasi-uniform entropy function $h_{QU}$ on a quasi-uniform space X, where h and $h_f$ are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map ψ of a $T_0$-quasi-uniform space $(X,U)$ we have $h\left(\psi \right)\le h_{QU}\left(\psi \right)$ when X is compact and $h_{QU}\left(\psi \right)\le h_f\left(\psi \right)$ with equality if X is a compact $T_2$ space.

The complement of the hyperplanes $\{x_i=x_j\}$, for all $i\ne j$, in $M^n$, where M is an aspherical 2-manifold, is known to be aspherical. Here we consider the situation when M is a 2-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical 2-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids.

We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.

We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.

In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.