Mark Mandelkern

On the uniform continuity of Tietze extensions

Tietze [5] proved the extension theorem for pointwise continuous functions on metric spaces, Urysohn [6] for normal topological spaces, and Kat&ov [3] for uniformly continuous functions on uniform spaces. Kat&ovs proof, using lemmata on binary relations, is quite complex. Isbells proof [2, p. 40] uses the Samuel compactification, thus traveling far outside the given uniform space. The purpose of this note is to show that, for metric spaces, a simple, direct, explicit and internal construction used many years ago for pointwise continuous functions will, in fact, produce a uniformly continuous extension.

A short proof of the Tietze-Urysohn Extension Theorem

Tietze [8] proved the extension theorem for metric spaces, and Urysohn I10] for normal topological spaces. Urysohn first proves his Lemma, which is a special case of the theorem. The proof of the lemma uses a set-theoretic argument which constructs a family of sets indexed by the rationals, and defines a continuous real-valued function using infima of subsets of the indices. In rather surprising contrast, the full extension theorem then makes use of infinite series, the Weierstrass M-test, and uniform convergence. The purpose of this note is to extend the method of Urysohns Lemma so as to obtain the extension theorem directly, without the use of uniform convergence, and without first proving the lemma. Urysohns Lemma itself is then no longer required, being an immediate corollary of the theorem.

Components of an open set

A classical theorem states that any open set on the real line is a countable union of disjoint open intervals. Here the numerical content of this theorem is investigated with the methods of constructive topology. 1980 Mathematics subject classification (Amer. Math. Soc.): 26 A 03, 54 A 99.

Constructive projective extension of an incidence plane

A standard procedure in classical projective geometry, using pencils of lines to extend an incidence plane to a projective plane, is examined from a constructive viewpoint. Brouwerian counterexamples reveal the limitations of traditional pencils. Generalized definitions are adopted to construct a projective extension. The main axioms of projective geometry are verified. The methods used are in accordance with Bishop-type modern constructivism.

Constructive irrational space

The Fréchet combination allows the construction of a complete metric on the set of irrational numbers. The constructive study of the resulting spaceM was begun by Errett Bishop. This paper studies the structure ofM in some detail. The constructive approach requires a strong form of the concept of irrational number and particular attention to the distinctions between the various notions of points exterior to a set. The main results are the characterization and construction of all compact and locally compact subspaces ofM.

Open subspaces of locally compact metric spaces

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.

Constructive harmonic conjugates

In the synthetic study of the real projective plane, harmonic conjugates have an essential role, with applications to projectivities, involutions, and polarity. The construction of a harmonic conjugate requires the selection of auxiliary elements; it must be verified, with an invariance theorem, that the result is independent of the choice of these auxiliary elements. A constructive proof of the invariance theorem is given here; the methods used follow principles put forth by Errett Bishop.

Finitary sequence spaces

This paper studies the metric structure of the space Hr of absolutely summable sequences of real numbers with at most r nonzero terms. Hr is complete, and is located and nowhere dense in the space of all absolutely summable sequences. Totally bounded and compact subspaces of Hr are characterized, and large classes of located, totally bounded, compact, and locally compact subspaces are constructed. The methods used are constructive in the strict sense. MSC: 03F65, 54E50.