Math. Maced. Vol. 8 (2010) 1-19 TOPOLOGICAL GAMES AND TOPOLOGIES ON GROUPS ALEXANDER V. ARHANGEL’SKII, MITROFAN M. CHOBAN AND PETAR S. KENDEROV Dedicated to Academician ´ Gor´ gi ˇ Cupona Abstract. In this paper, using the language and techniques of topological games, of sieves and of plumages, we give a conditions for a semitopologi- cal group or a paratopological group to be a topological group. We prove that a paratopological group with the Baire property and with given point- wise property is a topological group if and only if it is p-embedded in some pseudocompact space with respective property. The case of n-ary groups is examined too. Some new open problems are formulated. 1. Introduction By a space we understand a regular topological T 1 -space. We use the terminol- ogy from [7, 21]. Let ω = {0, 1, 2,...} and N = {1, 2,...}. By cl X H we denote the closure of a set H in a space X.A paratopological group is a group endowed with a topology such that the multiplication is jointly continuous. Recall that a semitopo- logical group is a group with a topology such that the multiplication is separately continuous. A semitopological group with a continuous inverse operation x → x −1 is called a quasitopological group. In 1936 D. Montgomery [26] has proved the following two theorems: Theorem 1M. Every completely metrizable separable semitopological group is a topological group. Theorem 2M. Every completely metrizable semitopological group is a paratopo- logical group. These two results of D.Montgomery have raised the following general problems: P1. What additional conditions are needed to be sure that a paratopological group is actually a topological group? P2. Under what additional conditions does a semitopological group become a paratopological or a topological group? 2000 Mathematics Subject Classification. 54H11, 54H20, 54H15. Key words and phrases. topological group, semitopological group, paratopological group, Baire property, fan-complete space, topological game, pseudocompact space. The second and the third author have been partially supported by the Bulgarian National Fund for Scientific Research, under grant DO02-360/2008. 1