On a theorem of Klee

Abstract

In 1953 and 1956, Klee [3], [4] proved that for E any infinitedimensional normed linear space and K any compact subset of E, E\K is homeomorphic to E. Klee's argument used sequences of bounded convex sets. In [5], Lin has some extensions of Klee's results using modifications of his methods. In this paper we give a short and elementary proof of a somewhat more general result' using only simple set-theoretic properties. A space S is said to be an a-space provided (1) S is an infinite-dimensional topological linear space, i.e., an infinite dimensional real vector space with a Hausdorff topology in which vector addition and scalar multiplication are jointly continuous, (2) S has a Schauder basis, i.e., a sequence {xi i>o of elements of S such that for each sES there is a unique sequence of scalars ai with s= aixi (convergence being in the topology of S) such that the function fi defined by fi(s) =ai is continuous for each i, and (3) there is a neighborhood U of the origin such that the elements xi I of the Schauder basis above are not in U. Henceforth, all spaces under discussion are to be a-spaces. For each i, let Mi denote the product of i copies of the reals with usual distance function di referring to distance between points, between a point and a set or between two sets. Let fi be as defined in condition (2) of the definition of an a-space and let gi be the map of S onto M, defined by gi(s) = (fi(s), f2(s), . . . , fi(s)). Since, by hypothesis, fi is continuous (for each i), then so is gj for each j. A set K CS is said to be projectible provided (1) K is closed, (2) for any pES\K, there is a j such that gj(p) is not an element of the closure of gj(K), and (3) there exist infnitely many i such that fi(K) is bounded above or below. The proof of the following lemma is trivial and is therefore omitted.