A Converse to Banach/s Contraction Theorem

Consider the hypothesis (H) that X admits a me tric p (yielding th e co rrect topo logy) s uch that I is a p-co ntrac tion. In a prev ious paper [1] 1 we gave co nditi ons on X s uch that f, if for some me tri c it sati sfied (1.1) locally at all points of X, would in fac t obey (H). No te that s uc h result s have a hypothes is whi c h is metric in nature. The prese nt paper is concern ed with topological conditions for (H), co nditi ons whose state me nts do not refer to any particular me tri c or me tri cs on X. In view of the Banach Contraction Theore m [2], two conditions which naturally s uggest the mselves are that , for some ~EX,