ABSTRACT
This chapter considers nearest points and farthest points in abstract spaces which are more general than normed linear spaces. It obtains results for subsets of these spaces to be proximinal, Chebyshev, remotal, and uniquely remotal, and also discusses singletonness of uniquely remotal sets. Approximation theory consists of the theory of nearest points and the theory of farthest points. The theory started with the concept of best approximation, which is concerned with the problem of describing the elements of a space X that may be approximated by the elements of a subset M of X. Most of the literature available in approximation theory is in spaces which are normed linear spaces. The absolute homogeneity of norm function, convexity of balls, and existence of non-trivial dual spaces of normed linear spaces have helped a lot in developing a fairly large theory of approximation in normed linear spaces.
