Resumen de Sharp Bases and Mappings

Haruto Ohta, Lei Mou

• Let S be the class of spaces with a sharp base in the sense of B. Alleche, A. V. Arhangel'skii and J. Calbrix (2000). A map f is called boundedly finite-to-one (resp. k-to-one) if there is a natural number k such that each fiber of f consists of at most (resp. exactly) k many points. Answering a question asked by C. Good, R. W. Knight and A. M. Mohamad (2002), we prove: (1) The image of a space in S under a perfect map or an open finite-to-one map is not necessarily in S, but every open boundedly finite-to-one image of a space in S is in S. (2) The preimage of a space in S under an open closed boundedly finite-to-one map is not necessarily in S, but every open k-to-one preimage of a space in S is in S.