Amalgamation-Type Properties of Arcs and Pseudo-Arcs
A continuum X is base if it satisfies the following “dual amalgamation” condition: whenever f : Y → X and g : Z → X are continuous maps from continua onto X, there is a continuum W and continuous surjections φ : W → Y , γ : W → Z such that f ◦ φ = g ◦ γ. A metrizable continuum is base metrizable if it satisfies the condition above, relativized to the subclass of metrizable continua. It is easy to show that simple closed curves are neither base nor base metrizable; however metrizable continua of span zero are known to be base metrizable. Furthermore, coexistentially closed continua are known to be base. The arc and the pseudo-arc are span zero; but, of the two, only the pseudo-arc is co-existentially closed. Hence the pseudo-arc is base metrizable for being span zero and base for being co-existentially closed. Here we show that: (i) there is a base metrizable continuum which is not span zero; and (ii) any metrizable continuum is base if and only if it is base metrizable.