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Explicit Metrization
Author(s) -
SHORE S. D.,
SAWYER LAURIE J.
Publication year - 1993
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1993.tb52535.x
Subject(s) - mathematical proof , metrization theorem , mathematics , simple (philosophy) , calculus (dental) , development (topology) , context (archaeology) , epistemology , computer science , pure mathematics , geometry , philosophy , mathematical analysis , geography , medicine , dentistry , separable space , archaeology
This paper provides a brief, nontraditional introduction to the historically important metrization theorems from 1910 to 1951. The intent is to show an evolution of ideas that lead to proofs for these results and that demonstrate how these theorems develop as a common thread. In this development it is suggested that the idea of distance function arises easily in the context of geometry and it is argued that this provides an avenue for developing the basic topological notions of convergence and neighborhood. In this setting strategies for constructing distance functions are provided and it is shown explicitly how these constructions can be used to give simple proofs of the classical theorems of Alexandroff and Urysohn (1923), Urysohn and Tychonoff (1925), Frink (1937), and Nagata, Smirnov, Bing (1951).