Title: Concerning Continuous Images of Rim-Metrizable Continua
Abstract: Mardesic ( 1962) proved that if X is a continuous, Hausdorff, infinite image of a compact ordered space K under a light mapping in the sense of ordering, then to(X) = to(K).He also proved (1967) that a continuous, Hausdorff image of a compact ordered space is rim-metrizable.Treybig (1964) proved that the product of two infinite nonmetrizable compact Hausdorff spaces cannot be a continuous image of a compact ordered space.We prove some analogues of these results for continuous Hausdorff images of rim-metrizable spaces.A topological space X is said to be rim-metrizable if X admits a basis of open sets whose boundaries are metrizable.A continuum is a compact connected Hausdorff space.Throughout the paper all spaces are assumed to be Hausdorff and all mappings are continuous.A compact ordered space admits a basis of open sets with at most two-point boundaries.Therefore, every compact ordered space is rim-metrizable.Since 1960 there has been a lot of work done on characterizing continuous Hausdorff images of compact ordered spaces.It is natural to ask whether these results can be generalized to continuous Hausdorff images of compact rim-metrizable spaces.Some of the results that have been obtained for continuous images of compact ordered spaces are the following:(1) In 1962 Mardesic [4] proved that if X is a continuous Hausdorff image of a compact ordered infinite space K under a light mapping in the sense of ordering, then the weight of X equals the weight of K.(2) In 1967 Mardesic [5] also proved that a continuous Hausdorff image of a compact ordered space is rim-metrizable.(3) In 1964 Treybig [6] proved that if the product of two infinite compact spaces X and Y is a continuous image of a compact ordered space, then X and Y must be metrizable.