Whitney continua of curves

A note on a Whitney map for continua

Let X be a non-metric continuum, and C(X) the hyperspace of subcontinua of X. It is known that there is no Whitney map on the hyperspace 2 for non-metrizable Hausdorff compact spaces X. On the other hand, there exist non-metrizable continua which admit and the ones which do not admit a Whitney map for C(X). In this paper we investigate the properties of non-metrizable continua which admit a Whi...

New Whitney-Type Formulae for Plane Curves

The classical Whitney formula relates the algebraic number of self-intersections of a generic plane curve to its winding number. We generalize it to an innnite family of identities, expressing the winding number in terms of the internal geometry of a plane curve. This enables us to split the Whit-ney formula by some characteristic of double points. It turns out, that only crossings of a very sp...

Center of Light Curves for Whitney Fold and Cusp

where Mag(x;y) = 1/| det[Jacη](x)| is the magnification of the lensed image x of y. Suppose that the light source at y moves along a path y(t). Then the shifted center-of-light curve relative to the source’s trajectory is defined by Xcl(t) = xcl(y(t)) − y(t). It is an important fact that locally stable k-plane lensing maps η are generic and each is differentiably equivalent about every critical...

The Planarity Theorems of MacLane and Whitney for Graph-like Continua

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane’s Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney’s Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

Topology Proceedings 1 (1976) pp. 209-219: SOME PROPERTIES OF WHITNEY CONTINUA IN THE HYPERSPACE C(X)