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On simultaneous strong proximinality

Gupta Sahil, Narang T.D.

Fasciculi Mathematici

2021

nr 65

47--55

In this paper, we extend the notions of simultaneous strong proximinality and simultaneous strong Chebyshevity available in Banach spaces to metric spaces and prove that if W is a simultaneously approximatively compact subset of a metric space (X, d) then W is simultaneously strongly proximinal. The converse holds if the set of all best simultaneous approximations to every bounded subset S of X from W is compact. We show that simultaneously strongly Chebyshev sets are precisely the sets which are simultaneously strongly proximinal and simultaneously Chebyshev. How simultaneous strong proximinality is transmitted to and from quotient spaces has also been discussed when the underlying spaces are metric linear spaces.