Vol. 49, No. 2, 1973

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On the lattice of proximities of Čech compatible with a given closure space

W. J. Thron and Richard Hawks Warren

Vol. 49 (1973), No. 2, 519–535

Let (X,c) be a Čech closure space. By M we denote the family of all proximities of Čech on X which induce c. M is known to be a complete lattice under set inclusion as ordering. The analogue of the R0 separation axiom as defined for topological spaces is introduced into closure spaces. R0-closure spaces are exactly those spaces for which Mϕ. Other characterizations for R0-closure spaces are presented. The most interesting one is: every R0-closure space is a subspace of a product of a certain number of copies of a fixed R0 closure space. A number of techniques for constructing elements of M are developed. By means of one of these constructions, all covers of any member of M can be obtained. Using these constructions the following structural properties of M are derived: M is strongly atomic, M is distributive, M has no antiatoms, |M| = 0,1 or |M|220.

Mathematical Subject Classification 2000
Primary: 54E05
Received: 21 June 1972
Published: 1 December 1973
W. J. Thron
Richard Hawks Warren