On irresolute multifunctions and related topological games

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.