Abstract:
For the universally measurable extension $C\rightarrowtail UM$ of the ring $C$ of continuous functions on a space $T$ the Gordon preimage $T\twoheadleftarrow g T$ is considered, which is the preimage of the maximal ideals of this extension. The new topological structure of Aleksandrov spaces with a cover and the concept of an enclosed covering of graduated type for these spaces are introduced. With the help of these concepts a topological characterization is given for the Gordon preimage $T\twoheadleftarrow gT$ as an enclosed covering of a certain type of space $T$ (Theorem 1). For comparison, a description of the hyper-Stonean preimage $T\twoheadleftarrow hT$ is presented without proof; the latter is the preimage of the maximal ideals of the Arens second dual extension $C\rightarrowtail C''$ (Theorem 2).
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