# 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.
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#### 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
• Mathematics
• 1984
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
• Mathematics
• 1997
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
• Mathematics
• 1999
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
• Mathematics
• 2004
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.)