$Top(X)$ within $\px$ ]{When lattices meet topology: $Top(X)$ within $\px$.}

Minimum Topological Group Topologies

A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.

Selection principles in function spaces with the compact-open topology

We continue to investigate applications of $k$-covers in function spaces with the compact-open topology.

Completeness and related properties of the graph topology on function spaces

The graph topology $τ_Γ$ is the topology on the space $C(X)$ of all continuous functions defined on a Tychonoff space $X$ inherited from the Vietoris topology on $X\times \mathbb R$ after identifying continuous functions with their graphs. It is shown that all completeness properties between complete metrizability and hereditary Baireness coincide for the graph topology if and only if $X$ is countably compact; however, the graph topology is $α$-favorable in the strong Choquet game, regardless of $X$. Analogous results are obtained for the fine topology on $C(X)$. Pseudocompleteness, along with…

有序紧致集的维氏拓扑适应

An Adaptation of the Vietoris Topology for Ordered Compact Sets

本文提出一种基于维氏拓扑的有序紧致集拓扑，通过比较不同乘积拓扑，展示了底空间的覆盖性质可能不传递到幂集的情况。

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu].

On the finite topology of a vector space and the domination problem for families of norms

Let $V$ be a real or complex vector space. The finite topology of $V$ consists of all the subsets $U$ for which the intersection $U \cap F$ is closed in $F$ for every finite-dimensional linear subspace of $V$. It is known that if $V$ has countable dimension, then this topology coincides with the greatest vector space topology and with the greatest locally convex vector space topology. Here, we note that in this case the finite topology is simply the union of all the normed space topologies. In connection to this problem, we characterize the vector spaces on which every family of norms with giv…

The C-compact-open topology on function spaces

This paper studies the C-compact-open topology on the set C(X) of all realvalued continuous functions on a Tychonov space X and compares this topology with several well-known and lesser known topologies. We investigate the properties C-compact-open topology on the set C(X) such as submetrizable, metrizable, separable and second countability.

On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$…

Metric topology on the moduli space

We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to the Tuschmann's question.

Set theory and topology. An introduction to the foundations of analysis. Part II: Topology - Fundamental notions

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as well as their applications to real numbers. Various methods to construct topological spaces are presented.

The Strongly Attached Point Topology of the Abstract Boundary For Space-Time

The abstract boundary construction of Scott and Szekeres provides a `boundary' for any n-dimensional, paracompact, connected, Hausdorff, smooth manifold. Singularities may then be defined as objects within this boundary. In a previous paper by the authors, a topology referred to as the attached point topology was defined for a manifold and its abstract boundary, thereby providing us with a description of how the abstract boundary is related to the underlying manifold. In this paper, a second topology, referred to as the strongly attached point topology, is presented for the abstract boundary c…

Strong topology on the set of persistence diagrams

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described. Also, we prove that the space of persistence diagrams with the bottleneck metric has infinite asymptotic dimension in the sense of Gromov.

Some notes on spaces of ideals and computable topology

We show some basic results on the characterization of quasi-Polish spaces in terms of spaces of ideals, with an emphasis on the connections with computable topology.

Recovering the boundary path space of a topological graph using pointless topology

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_1$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

通过开-闭弦拓扑研究拉格朗日子流形的拓扑

The topology of Lagrangian submanifolds via open-closed string topology

本文通过开-闭弦拓扑方法研究标准辛向量空间中拉格朗日子流形的拓扑，构造了链上dg代数的变形，并证明当π₂(L)=0时，L的Maslov类非消失。