On a Problem of Simultaneous Quasi-Uniform Extension

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Let T be a topology on a set X and X = ∪1n Xi, Xi ∩ Xj = 0 (i ≠ j), Ui a quasi-uniformity on Xi (i = 1,...,n). The paper establishes necessary and sufficient conditions for the existence of a quasi-uniformity U on X with the property that the topology induced by U coincides with T and the restriction of U to Xi is equal to Ui. The rather complicated condition in the general case can be simplified in the special case n = 2.