Handbook of Set-Theoretic Topology

Handbook of Set-Theoretic Topology

1984, Pages 169, 171-200
Handbook of Set-Theoretic Topology

CHAPTER 4 - Box Products

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Publisher Summary

This chapter discusses box products. The topology first considered on the set product of infinitely many spaces is that which is currently named the box topology. As this topology proved inefficient at preserving complex properties central to the field, it was soon replaced by the successful topology of Tychonov. The discovery that box topologies could provide the site for important counterexamples engendered today's scrutiny of the subject. The box products are never metrizable and often have a non-normal subspace. A box product of a small family of realcompact spaces is realcompact, but the box product of a countable family of compact spaces need not be normal. Within the class of topologically complete spaces are the realcompact spaces. A complete uniform space is the richest general structure preserved by box products. Every paracompact compact space is the topological sum of σ-compact spaces, and the topological sum of paracompact spaces is paracompact.

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