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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

1. Engelking, R., Dimension theory (Warszawa, 1978).Google Scholar
2. Efremovic, V. A., Infinitesimal spaces, Soviet Math, Doklady 76 (1951), 341–343.Google Scholar
3. Efremovic, V. A., The geometry of proximity spaces, Math, Sb 31 (1952), 89–200.Google Scholar
4. Naimpally, S. A. and Warrack, B. D., Proximity spaces (Cambridge University Press, 1970).Google Scholar
5. Smirnov, Ju. M., On proximity spaces, Amer. Math. Soc. Transi. Ser 2, 38 (1964), 4–35.Google Scholar
6. Smirnov, Ju. M., On the dimension of proximity spaces, Amer Math. Soc. Transi., Ser 2, 21 (1962), 1–20.Google Scholar
7. Smirnov, Ju. M., On dimension of remainder of compact extension of proximity and topological spaces, Math. Sb 69 (1966), 141–159.Google Scholar