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On Large Inductive Dimension of Proximity Spaces

Published online by Cambridge University Press:  20 November 2018

A. Kandil
A. Kandil*
Affiliation:
Mansoura University Mansoura, Egypt
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The notion of proximity spaces was introduced by Efremovic in [2, 3]. An analysis of proximity spaces was carried out by Smirnov in [5].

The study of covering dimension of proximity spaces was originated by Smirnov in [6].

In this paper we introduce the concept of δ-large inductive dimension of proximity spaces and study some of its properties.

1. Definitions and basic concepts.

Definition 1. [5]A proximity space or (δ-space) is a pair (X, δ) where X is a set and δ is a mapping from 2X × 2X into the set {0, 1} satisfying the following axioms:

  • 1. δ(A, B) = δ(B, A)∀ A, B ∊ 2X.

  • 2. δ(A, B ∪ C) = δ(A, B) δ(A, C) ∀ A, B, C ∊ 2X

  • 3. δ({x}, {y}) = 0 ⇔ x = y.

  • 4. δ(X, ∅) = 1.

  • 5. δ(A, B) = 1 ⇒ ∃ C, D ∊ 2X ∋ C ∪ D = X and δ(A, C) · δ(B, C) = 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 961 - 973
Copyright
Copyright © Canadian Mathematical Society 1983

References

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