Concerning Continuous Images of Rim-Metrizable Continua

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 $\omega (X) = \omega (K)$. 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.