Sam B. Nadler

Cones that are cells, and an application to hyperspaces☆

Abstract Let Y be a compact metric space that is not an ( n −1)-sphere. If the cone over Y is an n-cell, then Y ×[0,1] is an n-cell; if n ≤4, then Y is an ( n −1)-cell. Examples are given to show that the converse of the first part is false (for n ≥5) and that the second part does not extend beyond n =4. An application concerning when hyperspaces of simple n-ods are cones over unique compacta is given, which answers a question of Charatonik.

A result about a selection problem of Michael

It is shown that a continuum that is an S4 space in the sense of Michael must be hereditarily decomposable. This improves known results, thereby providing more evidence that such continua must be dendrites.

DEFINITIONS AND THEIR MOTIVATION: CONTINUITY AND LIMITS

ABSTRACT Many fundamental concepts are defined in textbooks by what I would call “working definitions.” This paper discusses the value of conceptual definitions and uses as illustrations the notions of continuity and limits as presented in elementary calculus courses. The paper thereby presents a different approach than the one used today for the introduction of these notions.

Maps between continua with stable values

Abstract Continua for which maps between them have stable values are studied. The case when the continua are Peano continua is emphasized.

Fixed point curves for linear interpolations of contraction maps

Let X be a banach space. If f0.f1 :X→X are contraction maps, then for each t in [0,1], let ft=(1−t)−f0+t−f1. This paper is concerned with the analytical properties of the curve of fixed points of the family |ft:0≤t≤1| and with the question of when, for a map G:[0,1]→X there are contraction maps g0,g1 : X→X such that gt[G(t)]=G(t) for all t in [0,1].