"While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavourer to write this book so that it is self-contained apart from requiring a knowledge of simple calculus, algebra and point set topology" Stephen Hawking & G.F.R. Ellis, The large scale structure of space-time. 1973 | "We show that [topology and computer science] are linked by the idea of convergence [. . . ] Point-set topology can be thought of as the study of convergence in general spaces." A. W. Roscoe, Topology and category theory in computer science, eds G.M. Reed, A.W. Roscoe, and R.F. Wachter. 1991 | "This book is a systematic exposition of a part of general topology that has proven useful in several branches of mathematics. I have, with difficulty, been prevented by my friends from labeling it: What Every Young Analysis Should Know." John L. Kelly, General Topology, 1955 |
The subject of topology is rather broad, and that makes it hard to give a useful, simple, short answer to the question "what is topology". Indeed, in Eric Weisstein's MathWord (the world's largest collection of mathematical definitions) topology is divided into twelve topics. In Dave Rusin's discussion of Geometric Areas of Mathematics, topology is divided into three areas according to the American Mathematical Society's Subject Classification Scheme: 54:general topology, 55: algebraic topology, and 57:manifolds, and these three are divided into several hundred topics. There is a topic in engineering called computational topology.
MacTutor History of Mathematics Archive (from University of St. Andrews, Scotland). See the article "Topology Enters Mathematics" under the History Topics Index, and the Short Biographies of Mathematicians. An interview (in Topology Atlas) with the renown Russian topologist Alexander V. Arhangel'skii includes some history of topology. Paraphrasing Arhangel'skii one might describe topology as "the science of infinite closeness with or without a concept of distance."
There is a home page for the Axiom of Choice , an axiom of set theory widely used in many areas of mathematics, including topology. See the humorous, accurate article by a feature columnist, Jim Holt. Axioms of set theory are part of the foundations of mathematics. Alexander Sakharov has compiled an organized collection of links concering the foundations of mathematics.
Topology Atlas is a huge data base of things related to topology, including a page of links to the question "What is topology", and a short electronic publication of mine.
More math links can be found on the home pages of most mathematicians. There is a list of home pages of topologists in Topology Atlas.
Go to my home page, the home page of the department, or to the UNCG home page.