A Pedagogical History of Compactness

@article{RamanSundstrm2015APH,
  title={A Pedagogical History of Compactness},
  author={Manya Raman-Sundstr{\"o}m},
  journal={The American Mathematical Monthly},
  year={2015},
  volume={122},
  pages={619 - 635}
}
Abstract This paper traces the history of compactness from the original motivating questions through the development of the definition to a characterization of compactness in terms of nets and filters. The goal of the article is to clarify the central concepts of open-cover and sequential compactness, including details that a standard textbook treatment tends to leave out. 
15 Citations

Topics from this paper

Simple way to prove compactness of closed intervals in simply ordered set with order topology
In this note, we present a simpler way to prove the compactness of the closed intervals in simply ordered set with order topology.
Introduction to Metric and Normed Spaces
In this chapter, we extend the notion of distance and absolute value from the real and complex number systems to more general spaces, in particular, spaces of functions.
Differentiation and Integration
In this chapter, we consider under what conditions and to what extent integration and differentiation are inverse operations on a function. We apply new results obtained by the author of this textExpand
General Measure Spaces
TLDR
In this chapter, results obtained for the real line are extended to more general spaces supplied with a measure. Expand
Measure on the Real Line
There are many examples of functions that associate a nonnegative real number or \(+\infty\) with a set. There is, for example, the number of members forming the set. Given a finite probabilityExpand
The Open Limit Point Compactness
In this paper, we gave a new topological concept and we called it the open limit point compactness.We have proved that each of the compactness, and the limit point compactness is stronger than of theExpand
Compactness on Soft Topological Ordered Spaces and Its Application on the Information System
It is well known every soft topological space induced from soft information system is soft compact. In this study, we integrate between soft compactness and partially ordered set to introduce newExpand
Open texture clarified
TLDR
‘Open texture’ is the property of concepts or terms that they are not fully defined with regard to unexpected questions, and it is shown that even though it is a ubiquitous phenomenon, there is no consensus on which way to go about defining it. Expand
Set Theory and Numbers
Set notation should be familiar to the reader. Recall x ∈ A means that x is an element of A; the negation is \(x\notin A\). Notation for every member of a set A belonging to a set B is \(A \subseteqExpand
Partial data querying through racing algorithms
TLDR
The main idea of the method is to identify the query that will be the most helpful in identifying the winning model in the competition, inspired from racing algorithms, which provides a general active learning technique that can be applied in principle to any model. Expand
...
1
2
...

References

SHOWING 1-10 OF 57 REFERENCES
E.W. Chittenden and the early history of general topology
Abstract E.W. Chittenden's work and its influence on the early history of general topology are examined. Particular attention is given to his work in metrization theory and its role in the backgroundExpand
The Rôle of Compactness in Analysis
(1960). The Role of Compactness in Analysis. The American Mathematical Monthly: Vol. 67, No. 6, pp. 499-516.
Historique de la notion de compacite
Abstract The Bolzano-Weierstrass and Borel-Lebesgue properties of sets constitute the fundamental ideas that led to the notion of compactness. The link between these ideas appeared for the first timeExpand
The establishment of functional analysis
Abstract This article surveys the evolution of functional analysis, from its origins to its establishment as an independent discipline around 1933. Its origins were closely connected with theExpand
Handbook of the History of General Topology
Introduction. Combinatorial Topology Versus Point-set Topology I.M. James. Elements of the History of Locale Theory P. Johnstone. Nonsymmetric Distances and their Associated Topologies: About theExpand
By their fruits ye shall know them: some remarks on the interaction of general topology with other areas of mathematics
In his letter of invitation to contribute to this “Handbook of the History of Topology”, Professor James asked us to discuss the role of general topology in other areas of topology. So this paper isExpand
The emergence of open sets, closed sets, and limit points in analysis and topology
Abstract General topology has its roots in real and complex analysis, which made important uses of the interrelated concepts of open set, of closed set, and of a limit point of a set. This articleExpand
History of functional analysis
Linear Differential Equations and the Sturm-Liouville Problem. The "Crypto-Integral" Equations. The Equation of Vibrating Membranes. The Idea of Infinite Dimension. The Crucial Years and theExpand
The Mathematical Works of Bernard Bolzano
PART I: GEOMETRY AND FOUNDATIONS 1.1 Elementary Geometry (1804) 1.2 A Better-Grounded Presentation of Mathematics (1810) PART II: EARLY ANALYSIS 2.1 The Binomial Theorem (1816) 2.2 A Purely AnalyticExpand
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
...
1
2
3
4
5
...