General Topology by John L. Kelley

Author:John L. Kelley
Language: eng
Format: epub
Publisher: INscribe Digital
Published: 2016-10-26T16:00:00+00:00

PRODUCTS OF COMPACT SPACES

The classical theorem of Tychonoff on the product of compact spaces is unquestionably the most useful theorem on compactness. It is probably the most important single theorem of general topology. This section is devoted to the Tychonoff theorem and a few of its consequences.

13 THEOREM (TYCHONOFF) The cartesian product of a collection of compact topological spaces is compact relative to the product topology.

PROOF Let where each Xa is a compact topological space and Q has the product topology. Let be the subbase for the product topology consisting of all sets of the form Pa–1[U] where Pa is the projection into the a-th coordinate space and U is open in Xa. In view of theorem 5.6 the space Q will be compact if each subfamily of , such that no finite subfamily of covers Q, fails to cover Q. For each index a let be the family of all open sets U in Xa such that Pa–1[U] ε . Then no finite subfamily of covers Xa and hence by compactness there is a point xa such that xa ε Xa ~ U for each U in . The point x whose a-th coordinate is xa then belongs to no member of and consequently is not a cover.

We give an alternate proof of Tychonoff’s theorem which does not depend on the Alexander theorem 5.6.

ALTERNATE PROOF (BOURBAKI) It will be proved that if is a family of subsets of the product and has the finite intersection property, then is not void. The class of all families which possess the finite intersection property is of finite character and consequently we may assume that is maximal with respect to this property by Tukey’s lemma 0.25(c). Because is maximal each set which contains a member of belongs to and the intersection of two members of belongs to . Moreover, if C intersects each member of , then C ε by maximality.* Finally, the family of projections of members of into a coordinate space Xa has the finite intersection property and it is therefore possible to choose a point xa in . The point x whose a-th coordinate is xa then has the property: each neighborhood U of xa intersects Pa[B] for every B in , or equivalen tly Pa–1[U] ε , for each neighborhood U of xa in Xa. Therefore finite intersections of sets of this form belong to . Then each neighborhood of x which belongs to the defining base for the product topology belongs to and hence intersects each member of . Therefore x belongs to B– for each B in , and the theorem is proved.

Several important applications of Tychonoff’s theorem occur in the chapter on function spaces; for the moment we consider a very simple consequence. A subset of a pseudo-metric space is bounded iff it is of finite diameter. Thus a subset of the space of real numbers is bounded iff it has both an upper and lower bound. The following is the classical theorem of Heine-Borel-Lebesgue.

14 THEOREM A subset of Euclidean n-space is compact if and only if it is closed and bounded.

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