On Open Extensions of Maps

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.