E. Michael

A quintuple quotient quest

Abstract This paper studies the images, under five kinds of quotient maps, of several standard types of topological spaces.

The product of a normal space and a metric space need not be normal

2. J. Nielson, Untersuchungen zur Topologie der geschlossenen Zweiseitigen Flëchen, Acta Math. 50 (1927), Satz 11, 266. 3. R. Baer, Isotopie von Kurven auf orientierbaren geschlossenen Fldchen una ihr Zusammenhang mit der topologischen Deformation der Flachen, Journ. f. Math. 159 (1928), 101. 4. W. Mangier, Die Klassen von topologischen Abbildungen einer geschlossenen Flache auf Sich, Math. Z. 44 (1938), 541, Satz 1, 2,542. 5. R. Fox, On the complementary domains of a certain pair of inequivalent knots, Nederl. Akad. Wetensch. Proc. Ser. A 55-Indag. Math. 14 (1952), 37-40. 6. L. Neuwirth, The algebraic determination of the topological type of the complement of a knot, Proc. Amer. Math. Soc. 12 (1961), 906.

A note on closed maps and compact sets

Some results relating closed maps to compact sets, which are already known for metrizable spaces, are here proved in a more general setting. Examples are given to indicate barriers to further improvements.

Separable Banach spaces which admitl n ∞ approximations

In this paper we study a class of separable Banach spaces which can be approximated by certain special finite-dimensional subspaces. This class is characterized in Theorem 1.1, from which it follows that the space of continuous scalar-valued functions on a compact metric space always belongs to this class, and that every member of this class has a monotone basis.

A selection theorem

2. -, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. 3. V. L. Klee, Convex bodies and periodic homeomorphism in Hilbert spaces, Trans. Amer. Math. Soc. 74 (1953), 10-43. 4. -, A note on topological properties of normed linear spaces, Proc. Amer. Math. Soc. 7 (1956), 673-674. 5. Bor-Luh Lin, Two topological problems concerning infinite-dimensional normed linear spaces, Trans. Amer. Math. Soc. 114 (1965), 156-175. 6. J. R. Retherford and C. W. McArthur, Some remarks on bases in linear topological spaces, Math. Ann. 64 (1966), 38-41.

Almost complete spaces, hypercomplete spaces and related mapping theorems

Abstract Thirty years ago, Zdenek Frolik introduced a class of spaces, subsequently called almost Cech- complete by Aarts and Lutzer, and showed that a completely regular space is in this class if and only if it has a dense Cech-complete subspace. This first purpose of this note is to obtain some new characterizations for these spaces, which are here simply called almost complete. The second purpose is to prove a number of mapping theorems, most of which are related to almost complete spaces and two of which sharpen recent results of Stegall and of Coban and Kenderov.

Compact-covering images of metric spaces

Several kinds of compact-covering images of metric spaces are characterized.

Continuous selections avoiding a set

Abstract Conditions are obtained under which a set-valued function ϕ : X ⇒ 2 Y has a continuous selection which avoids a set E ⊏ Y .

A generalization of a theorem on continuous selections

The authors selection theorem for functions with finite-dimensional domain is sharpened in two directions.

On representing spaces as images of metrizable and related spaces

Abstract The following question is studied. Suppose Y is a space with a covering . When and how can Y be expressed as the image of a metrizable space, or of a space admitting a perfect (or quasi-perfect) map onto a metrizable space, under a mapping with properties related to those of the covering ?