A STRONG COMPLETENESS CONDITION IN UNIFORM SPACES WITH WELL ORDERED BASES
A natural strengthening of the completeness hypothesis, called -completeness, allows us to extend to ! -metric spaces some properties of complete metric spaces which fail to hold if only the completeness condi- tion is assumed. In this way we achieve some results on hyperspace theory, dealing with supercompleteness and with selections of multivalued functions. The notion of -completeness is investigated and it is proved that quite important classes of ! -metric spaces have this property.
On closed images of the space of irrationals
In connection with his recent joint work with A. H. Stone on quotients of the space of irrationals (cf. [6]), E. Michael asked whether every complete separable metric space is a continuous image of the irrationals under a closed mapping. The present note contains the positive answer to this question. In fact, we actually prove a more general result which does not assume separability. The proof is based on a refinement of a known lemma (see [5, p. 281], and [2] or [4] for the nonseparable case) that every nonempty closed subset of a metric space X of covering dimension 0 is a retract of X. The refinement, which seems to be new even in the separable case, asserts the existence of a closed retraction. Our construction of such a retraction coincides with one given in [9, Theorem 3', p. 8] by A. H. Stone, who did not, however, observe that it is closed. We include the details for the sake of completeness. I am grateful to E. Michael for some valuable remarks.
Absolute neighborhood retracts and local connectedness in arbitrary metric spaces
© Foundation Compositio Mathematica, 1956-1958, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.