Introduction to Topology by Bert Mendelson

Author:Bert Mendelson
Language: eng
Format: epub
Publisher: Dover Publications

Since this set is an open subset of X the projection maps are continuous.

A subset O1 × O2 × . . . × On of X can be written as so that we have a guide to the appropriate topology on an arbitrary product of topological spaces.

DEFINITION 7.6Let be an indexed family of topological spaces. The topological product of this family is the set with the topology consisting of all unions of sets of the form

We have used as a basis for the topology the collection of sets of the form . That is a topology follows from the fact that , and a finite intersection of elements of is again in . Clearly this topology makes the projection maps continuous. Since any topology on X which makes the projection maps continuous must contain the sets of this form, the product topology is the weakest topology consistent with the continuity of the projection maps.

It is easily seen that, analogous to Proposition 7.4, a basis for the neighborhoods at a point x is the collection of sets of the form , where Nαi is a neighborhood of . In effect then, in the product space X we are saying that a point y is in a given neighborhood of x or is close to x if there is a finite set of indices {α1, . . . , αk} such that y(αi) is close to x(αi).

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