Department of Topology hyperspaces upper semi-finite topology HS-spaces F-normal spaces K-, K’-, K’’- and K*-spaces continuous maps closed maps T2-, T3- and T4-spaces
Institute of Mathematics with Computer Center at the Bulgarian Academy of Sciences
On the basis of a property of the hyperspace, H.-J. Schmidt [S] introduced implicitly a class of spaces called in the present paper HS-spaces. In [S, Theorem 11(3d)] it is stated that every Hausdorff HS-space is regular. Later on M. Paoli and E. Ripoli noted in [PR1] that the proof of the above assertion is incorrect, but the question of the correctness of the statement is open. We give a partial solution of this problem introducing a large class of spaces, which contains all Hausdorff spaces with countable character, where the theorem holds. Two reduction theorems are obtained as well. 1991 AMS Subj. Class.: primary – 54B20, 54CO5, 54D10, 54D15; secondary – 54B05, 54C10, 54G99.