Mapping theorems on mesocompact spaces

Kuo Shih Kao, Li Sheng Wu
1983 Proceedings of the American Mathematical Society  
In this paper we prove two mapping theorems on mesocompact spaces: (1) Perfect mappings preserve mesocompactness; (2) Closed mappings preserve normal mesocompactness. The main results of this paper are two mapping theorems on mesocompact spaces. Mesocompactness was defined in J. R. Boone [4] and studied by V. J. Mancuso [10] and J. R. Boone [4, 5]. Mancuso [10] intended to prove that perfect mappings preserve mesocompactness, but his proof was incorrect. J. R. Boone [5] noticed the error in
more » ... ed the error in Mancuso's proof but he gave a proof only for a special case (the domains of the mappings were assumed to be normal). Our Theorem 1 solves the Mancuso problem. Boone [6] studied /c-quotient mappings and proved that /c-quotient, closed mappings preserve normal mesocompactness. Our Theorem 2 improves the foregoing result by omitting the condition "/c-quotient" in the statement. In this paper, normal spaces are assumed to be Tx, and all mappings are continuous and surjective. Let % be a collection of subsets of X, the union
doi:10.1090/s0002-9939-1983-0712651-5 fatcat:uxf2byqgdbe3xbu5ic7r2ngrga