  This gives two existence results of alpha-core solutions by introducing
P-open conditions and strong P-open conditions into games without ordered
preferences. The existence of alpha-core solutions is obtained for games with
infinite-players. Secondly, it provides a short proof of Kajii's (Journal of
Economic Theory 56, 194-205, 1992) existence theorem for alpha-core solutions,
further, the Kajii's theorem is equivalent to the Browder fixed point theorem.
In addition, the obtained existence results can include many typical results
for alpha-core solutions and some recent existence results as special cases.
