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2 - General Topology

Published online by Cambridge University Press:  06 July 2010

R. M. Dudley
R. M. Dudley
Affiliation:
Massachusetts Institute of Technology
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Summary

General topology has to do with, among other things, notions of convergence. Given a sequence xn of points in a set X, convergence of xn to a point x can be defined in different ways. One of the main ways is by a metric, or distance d, which is nonnegative and real-valued, with xn → x meaning d(xn, x) → 0. The usual metric for real numbers is d(x, y) = ∣x - y∣. For the usual convergence of real numbers, a function f is called continuous if whenever xn → x in its domain, we have f(xn) → f(x).

On the other hand, some interesting kinds of convergence are not defined by metrics: if we define convergence of a sequence of functions fn “pointwise,” so that fn → f means fn(x) → f(x) for all x, it turns out that (for a large enough class of functions defined on an uncountable set) there may be no metric e such that fn→ f is equivalent to e(fn, f) → 0.

Given a sense of convergence, we can call a set F closed if whenever xi ∈ F for all i and xi → x we have x ∈ F also. Any closed interval [a, b] ≔ x: a ≤ x ≤ b is an example of a closed set. The properties of closed sets F and their complements U ≔ X∖F, which are called open sets, turn out to provide the best and most accepted way of extending the notions of convergence, continuity, and so forth to nonmetric situations.

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Real Analysis and Probability , pp. 24 - 84
Publisher: Cambridge University Press
Print publication year: 2002

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  • General Topology
  • R. M. Dudley, Massachusetts Institute of Technology
  • Book: Real Analysis and Probability
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755347.003
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  • General Topology
  • R. M. Dudley, Massachusetts Institute of Technology
  • Book: Real Analysis and Probability
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755347.003
Available formats
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  • General Topology
  • R. M. Dudley, Massachusetts Institute of Technology
  • Book: Real Analysis and Probability
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755347.003
Available formats
×