Topology and its Applications最新文献

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. It is also well known that the mapping class group $\mathrm{Mod}(N_g;k)$ of a non-orientable surface can be identified with a subgroup of $\mathrm{Mod}(S_{g-1};2k)$, the mapping class group of its orientable double cover. These facts, together with the classical Nielsen realization theorem, are used to prove that every finite subgroup of $\mathrm{Mod}(N_g;k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\mathrm{Diff}(N_g;k)$. In contrast, we show the projection $\mathrm{Diff}\left(N_g\right)\to \mathrm{Mod}\left(N_g\right)$ does not admit a section for large g.

In this paper, we introduce polytopal k-wedge construction and blowdown of a simple polytope and inspect the effect on the retraction sequence of a simple polytope due to the k-wedge construction and blowdown. In relation to these constructions, we introduce the k-wedge and blowdown of a quasitoric orbifold. We compare the torsions in the integral cohomologies of k-wedges and blowdowns of a quasitoric orbifold with the original one. These two constructions provide infinitely many integrally equivariantly formal quasitoric orbifolds from a given one.

We will calculate the density, the spread and related cardinal functions on lexicographic products of GO-spaces, and give their applications.

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Well-known classes of groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods of the diagonal are used as a means of study.

The theory of generalized metrizable spaces is an important topic of general topology. This paper is a survey of research methods and achievements on generalized metrizable properties in topological groups and weakly topological groups. We mainly study this kind of properties in topological groups, semitopological groups, paratopological groups, quasitopological groups and free topological groups, and focus on the influence of separation properties, conditions for weakly topological groups to become topological groups, cardinal invariants, weak first-countability, three-space properties and remainders in compactifications on topological groups and related structures. Finally, some unsolved problems in this field are listed for researchers.

In this paper, we introduce the notion of a semi-stratifiable frame as an extension of classical semi-stratifiability and also as the monotonization of perfect frames. We show that stratifiable frames are precisely the monotonically normal semi-stratifiable frames. Moreover, we present an insertion theorem for semi-stratifiable frames in terms of real functions and thereby obtain an insertion theorem for stratifiable frames.

We prove, in $\mathrm{ZFC}$, that if $G$ is an infinite countable Abelian group, then there is an ultrafilter $p\in {\omega }^{⁎}$ such that $\left({\mathrm{Ult}}_p\right(G),{\tau }_{\overline{\text{Bohr}}})$ has non-trivial convergent sequences, consequently $\left({\mathrm{Ult}}_p^{{\omega }_1}\right(G),{\tau }_{\overline{\text{Bohr}}})$ has non-trivial convergent sequences, extending Theorem 3.9 from [13]. In addition, we prove that the Remark 3.8 from [13] is false; so, the proof of the Corollary 3.11 is false too.

In this paper, the notion of an original point for a topological space is introduced. Then characterizations of some spaces such as stratifiable spaces and MCP-spaces in terms of set-valued maps with values in $F\left(Y\right)$, where Y has an original point, are presented.

In this paper we continue researches deal with increasing sequence $\left\{T_m\right\}$ of $T_0$-topologies on the set of positive integers focusing on the properties of arithmetic progressions in topologies $T_m$. We characterize the closures of arithmetic progressions in all topologies $T_m$ on $N$. Additionally, we present the characterization of the closures of arithmetic progressions in the common division topology $T$ on $N$. Moreover, for each $m\in N$ we characterize regular open arithmetic progressions in $(N,T_m)$ and we examine which of these spaces are semiregular.

This paper continues a program due to Motegi regarding universal bounds for the number of nonisotopic essential n-punctured tori in the complement of a hyperbolic knot in $S^3$. For $n=1$, Valdez-Sánchez showed that there are at most five nonisotopic Seifert tori in the exterior of a hyperbolic knot. In this paper, we address the case $n=2$. We show that there are at most six nonisotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.