The Origins of Set-Theoretic Topology
Overview
Topology was a generalization of geometry that began to assume a separate identity in the nineteenth century. The study of set theory was first made a serious part of mathematics at the end of the nineteenth century. The two were combined in the twentieth century to the advantage of both, as set theory (which until then had seemed to have little to do with ordinary mathematics) found an application in topology and topology (which was capable of wandering off in many directions from geometry) could have a place of its own with the help of set theory. Many of the traditional areas of mathematics could be restated in general terms with the help of set-theoretic topology, which gave a new range of application to terms like "function."
Background
Geometry was one of the earliest branches of mathematics to assume identity as a separate discipline, largely thanks to the efforts of Euclid (fl. c. 300 B.C.) in his work the Elements. Euclid remained the source for all matters geometric well into the nineteenth century, although some by that time had come to worry about the specific form that the axioms (the assumptions from which Euclid started) took.
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