id: 06528703
dt: j
an: 2016c.00018
au: Raman-Sundström, Manya
ti: A pedagogical history of compactness.
so: Am. Math. Mon. 122, No. 7, 619-635 (2015).
py: 2015
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: A30 I95
ut: compactness; sequential compactness; countable compactness
ci: Zbl 0951.54001
li: doi:10.4169/amer.math.monthly.122.7.619
ab: This nice paper is “an attempt to fill in some of the information that
the standard textbook treatment of compactness leaves out. It is not a
historical article, {\it per se}, but a synthesis of historical
documents with an eye towards clarifying the main ideas related to
compactness.” Arguing about the ideas that motivated the notion of
compactness, the author discusses the influence of the study of
properties of closed, bounded intervals of real numbers (Weierstrass,
Heine-Borel and Cousin theorems), spaces of continuous functions
(Arzelà-Ascoli theorem and a criterion of compactness of subsets of
$C^{0}[a,b]$), and solutions to differential equations (Peano existence
theorem). The development of the two central concepts of compactness
stemming from sequences (Bolzano-Weierstrass property) and open covers
of real numbers (Borel-Lebesgue property) is traced through the
analysis of the notions of countable and limit points compactness
(Fréchet), compactness on metric spaces (Hausdorff), and open-cover
compactness (Alexandroff and Urysohn). The paper concludes with an
overview of the theory of nets (Moore and Smith) and filters (Cartan
and Smith); both being included in the latest edition of a popular
textbook by {\it J. R. Munkres} [Topology. 2nd ed. Upper Saddle River,
NJ: Prentice Hall (2000; Zbl 0951.54001)].
rv: Svitlana P. Rogovchenko (Kristiansand)