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Nonconstant continuous maps of spaces and of their β-compactifications

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
33
Issue
1
Identifiers
DOI: 10.1016/0166-8641(89)90087-4
Keywords
  • Representation Of Monoids
  • β-Compactification
  • Continuous Maps

Abstract

Abstract For every cardinal number ɱ and every pair of monoids M 1 ⊆ M 2 there exists a Tychonoff space X such that the collection of all subspaces Y of β X with X ⊆ Y ⊆ βX contains a stiff collection X of the cardinality ɱ (stiff in the sense that if Y, Y′ ∈ X and ƒ: Y → Y′ is continuous, then either ƒ is constant or Y = Y′ and ƒ is the identity); moreover, all the nonconstant continuous maps of X into itself form a monoid isomorphic to M 1 and all the nonconstant continuous maps of β X into itself form a monoid isomorphic to M 2. This assertion is a corollary of the Main Theorem proved here. A more general setting of simultaneous representations of small categories is investigated.

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