Fans whose hyperspaces are cones.
by
Sergio Macias
Universidad Nacional Autonoma de Mexico

A continuum is a compact connected metric space. A fan is an arcwise connected continuum such that the intersection of any two subcontinua is connected and has exactly one point which is the common part of three otherwise disjoint arcs call the top of the fan.

Given a continuum X, we define its hyperspaces as the following sets:

2X={A subset X | A is closed and nonempty}

Cn(X)={A in 2X | A has at most n components}

Fn(X)={A in 2X | A has at most n points}

It is known that 2^X is a metric space with the Hausdorff metric.

Given a fan F, let G(F) denote any of the hyperspaces 2^F, C_n(F) and F_n(F), for n >= 2.

In this talk we present the following resut:

Theorem. If F is a fan with top \tau, which is homeomorphic to the cone over a compact metric space, then G(F) is homeomorphic to the cone over a continuum.

Date received: August 5, 2002