On irresolute multifunctions and related topological games

Sewalem Ghanem; Abdelfattah A. El Atik; Sewalem Ghanem; Abdelfattah A. El Atik

doi:10.3934/math.20221026

AIMS Mathematics

2022, Volume 7, Issue 10: 18662-18674. doi: 10.3934/math.20221026

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Research article

On irresolute multifunctions and related topological games

Sewalem Ghanem ^1,2,
Abdelfattah A. El Atik ^{2
,
,}

1.
Department of Mathematics, College of Science and Arts, Qassim University, Alnbhaniah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

Received: 05 May 2022 Revised: 01 August 2022 Accepted: 04 August 2022 Published: 22 August 2022
MSC : 54A05, 54B10, 54D30, 54G99

In this paper, we introduce and study $ \alpha $-irresolute multifunctions, and some of their properties are studied. The properties of $ \alpha $-compactness and $ \alpha $-normality under upper $ \alpha $-irresolute multifunctions are topological properties. Also, we prove that the composition of two upper and lower $ \alpha $-irresolute multifunctions is $ \alpha $-irresolute. We apply the results of $ \alpha $-irresolute multifunctions to topological games. Upper and lower topological games are introduced. The set of places for player ONE in upper topological games may guarantee a gain is semi-closed. Finally, some optimal strategies for topological games are defined and studied.
- multifunctions,
- winning strategy,
- topological games,
- upper and lower irresolute multifunctions
Citation: Sewalem Ghanem, Abdelfattah A. El Atik. On irresolute multifunctions and related topological games[J]. AIMS Mathematics, 2022, 7(10): 18662-18674. doi: 10.3934/math.20221026

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Abstract

In this paper, we introduce and study $ \alpha $-irresolute multifunctions, and some of their properties are studied. The properties of $ \alpha $-compactness and $ \alpha $-normality under upper $ \alpha $-irresolute multifunctions are topological properties. Also, we prove that the composition of two upper and lower $ \alpha $-irresolute multifunctions is $ \alpha $-irresolute. We apply the results of $ \alpha $-irresolute multifunctions to topological games. Upper and lower topological games are introduced. The set of places for player ONE in upper topological games may guarantee a gain is semi-closed. Finally, some optimal strategies for topological games are defined and studied.

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