A universal mapping theorem for the hyperspace suspension
by
Raúl Escobedo
Universidad Autónoma de Puebla
Coauthors: María de Jesús López and Sergio Macías

Let X be a continuum (a nonempty compact connected metric space). Let C(X) denote the hyperspace of all subcontinua of X, with the Hausdorff metric. Let F₁(X) denote the space of singletons of X. The Hyperspace Suspension of X, denoted by HS(X), is the quotient space C(X)/F₁(X).

Each mapping f\colon X --> Y, between continua, induces a mapping between the corresponding hyperspaces suspension, HS(f)\colon HS(X) --> HS(Y). A mapping f\colo X --> Y, between continua, is said to be universal, provided that for each mapping g\colon X --> Y, there exists a point x in X such that g(x)=f(x).

We say that a continuum X has zero span provided that for each continuum Z in X×X such that \pi₁(Z)=\pi₂(Z) we have that Z \cap \Delta =/= \emptyset where \Delta is the diagonal in X×X.

We show the following:

Theorem. Let f\colon X --> Y be an onto mapping between continua, where Y is a continuum with zero span. Then the induced mapping HS(f)\colon HS(X) --> HS(Y) is universal.

Date received: February 12, 2002