Completeness

Abstract

A non-closed subspace of a topological space is singularly defective, particularly if the topology determines geometric structure. We therefore present completeness as a property of universal closure, confining our attention to metric spaces. We shall see that extraordinarily powerful theorems are available in a metric space which contains every possible boundary point in every possible metric superspace.

But I—that am not shap’ d for sportive tricks, Nor made to court an amorous looking-glass— I—that am rudely stamp’d, and want love’s majesty To strut before a wanton ambling nymph— I—that am curtail ’d of this fair proportion, Cheated of feature by dissembling nature, Deform ’d, unfinish ’d, sent before my time Into this breathing world, scarce half made up ... Richard III, l,i.