Every non-normable Frechet space is homeomorphic with all of its closed convex bodies

Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications

“…This result was generalized to every Banach space with the help of Bessaga's non-complete norm technique (see the book by Bessaga and Peĺczyński [7]). To get a better insight in the history of the topological classification of convex bodies the reader should also look at the papers by Stocker [25], Corson and Klee [10], and Bessaga and Klee [5,6]. In [15], T. Dobrowolski gave a C p smooth version of that result which held within the class of WCG Banach spaces.…”

Topological Properties of the Hilbert Cube and the Infinite Product of Open Intervals

“…Let hi be defined so that (1) We wish to verify that, in fact, the left product of {A(}(>0 is a /3*-homeomorphism satisfying the conditions of the theorem. Since with r= 1, the conditions of Lemma 2.2 with respect to r, are clearly satisfied using particularly conditions (1), (2), and (5) above, and since conditions (3) and (4) imply that the other hypotheses of Lemma 2.2 are satisfied, then the left product of {A¡}i>0 is a mapping A of 7™ onto itself. To verify that A is a homeomorphism it suffices to note that no two points p and q can be mapped to the same point by h. From conditions (5) and (6) it follows that for some n and all m > n rX(hn-■ -hy(p)) = ry(hm-■ ■ hy(p)) = ry(h(p)) and ri(A"-• hy(q)) = ry(hm-■ hy(q)) = ry(h(q)).…”

“…Let {£i}i>0 be any collection of closed sets such that each is weakly thin with respect to a subpartition whose elements are disjoint from a. Then there exists a ß-homeomorphism h such that (1) for p e (°7»\Ui>o Kt), h(p) e °7<°, (2) for any i>0, there is ajt such that h(Ky)^ Wji and (3) a(h) <=«'.…”

“…The author [1] has the same conclusion (not only for compact sets, but for certain noncompact sets which are appropriately imbedded) for a class of topological linear spaces which does not appear to include all normed spaces and does not include s but does include certain nonnormable spaces. In [2] Bessaga and Klee establish that the removal of a single point from any of a very general class of topological linear spaces including s does not change the topological character of the space.…”

Products of complexes and Fréchet spaces which are manifolds

“…With B denoting the unit ball of E, it is sufficient to demonstrate that K x YiieN Bi is a n¡eN 7írmanifold, for by a theorem of Bessaga and Klee [6] (see note added in proof), each infinite-dimensional Fréchet space is homeomorphic to each of its closed convex bodies. Thus, E is homeomorphic to B and to YJieN Eu so it is homeomorphic to TIíen B¡.…”