The Compactness of the Hyperspace 2X with the Locally Finite Topology

Let PR(X) denote the Pixley-Roy topology on the collection of all nonempty, finite subsets of a space X. For each cardinal κ, let κ* be the cardinal κ with the co-finite topology. We use PR(κ*) to obtain a partial solution in ZFC to F. Tall's question whether every normal, locally compact, metacompact space is paracompact [6]. W.S. Watson has answered this question affirmatively assuming V = L[7]. The question also has an affirmative answer if we assume either that the space is perfectly normal [1] or that it is locally connected [4].A space X is said to be boundedly metacompact (boundedly paracompact) provided that for each open cover of X there is a positive integer n such that has a point finite (locally finite) open refinement of order n. As the main result of this paper, we show every normal, locally compact, boundedly metacompact space is paracompact.

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Locally Finite Groups

10.1016/s0924-6509(08)x7028-x ◽

1973 ◽

Keyword(s):

Finite Groups ◽

Locally Finite ◽

Locally Finite Groups

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Examples of theories of presheaf type

10.1093/oso/9780198758914.003.0011 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

Finite Groups ◽

Linear Independence ◽

Geometric Theory ◽

Vector Spaces ◽

Linear Orders ◽

Locally Finite ◽

Strong Unit ◽

Locally Finite Groups ◽

Simplicial Sets ◽

Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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Existentially closed central extensions of locally finite p-groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066093 ◽

1986 ◽

Vol 100 (2) ◽

pp. 281-301 ◽

Cited By ~ 5

Author(s):

Felix Leinen ◽

Richard E. Phillips

Keyword(s):

Central Extensions ◽

Locally Finite ◽

Existentially Closed ◽

Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

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