Nonstandard methods in large-scale topology II

Takuma Imamura

英文摘要：

This paper is a sequel of Imamura (2019) (arXiv:1711.01609) where we set up a framework of nonstandard large-scale topology. In the present paper, we apply our framework to various topics in large-scale topology: spaces having with both small-scale and large-scale structures, large-scale structures on nonstandard extensions, size properties of subsets of coarse spaces, and coarse hyperspaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Fehrele's principle in nonstandard topology

In nonstandard analysis, Fehrele's principle is a beautiful criterion for a set to be internal, stating that every galactic halic set is internal. In this note, we use this principle to prove some well-known results in topology, including slight generalisations of the Moore-Osgood theorem and Dini's theorem.

2020-10-06math.GN

Nonstandard Analysis in Topology

We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.

2011-07-17math.GN

Nonstandard Topology without Logic, Ultrafilters as infinitesimal points in topological space

Nonstandard analysis is very complex, so finding a simple description of infinitesimal points will be useful. In this paper, ultrafilters as infinitesimal points in a topological space will be proposed, and some topological concepts is restated by this tools.

2013-02-13math.GN

Nonstandard methods in large-scale topology

We develop some nonstandard techniques for bornological and coarse spaces. We first generalise the notion of bornology to prebornology, which better fits to coarse spaces. We then give nonstandard characterisations of some basic large-scale notions in terms of galaxies and finite closeness relations, concepts that have been developed for metric spaces. Some hybrid notions that involve both small-scale and large-scale are also discussed. Finally we illustrate an application of our nonstandard characterisations to prove some elementary facts in large-scale topology and functional analysis, e.g.,…

2019-06-23math.GN

Topologizing Rings of Map Germs: An Order Theoretic Analysis of Germs via Nonstandard Methods

Using nonstandard analysis we define a topology on the ring of germs of functions: $(mathbb R^n,0)\rightarrow(mathbb R,0)$. We prove that this topology is absolutely convex, Hausdorff, that convergent nets of continuous germs have continuous germs as limits and that, for continuous germs, ring operations and compositions are continuous. This topology is not first countable, and, in fact, we prove that no good first countable topology exists. We give a spectrum of standard working descriptions for this topology. Finally, we identify this topological ring as a generalized metric space and examin…

2012-06-03math.GN

Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications

\begin{abstrac} Let $(X,T) $ be a topological space, and $^{*}X$ a non--standard extension of $X$. There is a natural ``standard'' topology $^{S}T$ on $^{*}X$ generated by $^{*}G$, where $G\in T$. The topological space $(^{*}X,^{S}T) $ will be used to study compactifications of $(X,T)$ in a systematic way.

2006-01-30math.GNmath.LO

更多相关论文收起更多论文

Compactness of $ω^λ$ for $λ$ singular

We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.

2013-04-07math.GNmath.LO