Author Topic: Topological Space  (Read 1855 times)

Offline Alberto Dominguez

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Topological Space
« on: December 18, 2011, 07:42:40 PM »
A topological space is a set X together with \tau (a collection of subsets of X) satisfying the following axioms:
The empty set and X are in \tau.
\tau is closed under arbitrary union.
\tau is closed under finite intersection.
The collection \tau is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. A topological space in which the points are functions is called a function space. The sets in \tau are called the open sets, and their complements in X are called closed sets.

Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.

« Last Edit: December 18, 2011, 07:52:30 PM by Alberto Dominguez »
Kind regards,

Alberto Domínguez
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