Multiset proximity spaces

Kandil, A.; Tantawy, O. A.; El-Sheikh, S. A.; Zakaria, Amr

doi:10.1016/j.joems.2015.12.002

Multiset proximity spaces

Authors

¹ Department of Mathematics, Faculty of Science, Helwan University, Helwan, Egypt

² Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

³ Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

10.1016/j.joems.2015.12.002

Abstract

A multiset is a collection of objects in which repetition of elements is essential. This paper
is an attempt to explore the theoretical aspects of multiset by extending the notions of compact,
proximity relation and proximal neighborhood to the multiset context. Examples of new multiset
topologies, open multiset cover, compact multiset and many identities involving the concept of multi-
set have been introduced. Further, an integral examples of multiset proximity relations are obtained.
A multiset topology induced by a multiset proximity relation on a multiset M has been presented.
Also the concept of multiset δ- neighborhood in the multiset proximity space which furnishes an
alternative approach to the study of multiset proximity spaces has been mentioned. Finally, some
results on this new approach have been obtained and one of the most important results is: every T 4 -
multiset space is semi-compatible with multiset proximity relation δ on M ( Theorem 5.10 ).

Keywords