Sergio Macías

On the hyperspaces Cn(X) of a continuum X

Abstract In 1939 M. Wojdyslawski showed that a continuum X is locally connected if and only if for each positive integer n , C n (X) is an absolute retract. Since then, nothing else has been done about these hyperspaces. We present some of the properties of these hyperspaces.

On symmetric products of continua

Abstract We show that the nth symmetric product of a continuum is unicoherent if n ⩾ 3. We prove that the arc is the only finite dimensional continuum for which its second symmetric product is homeomorphic to its hyperspace of subcontinua, and that for any finite dimensional continuum X its hyperspace of subcontinua is not homeomorphic to its nth symmetric product for any n ⩾ 3.

Hyperspaces and cones

We characterize locally connected continua X for which its hyperspace of subcontinua, C(X), has finite dimension and is homeomorphic to the cone of a continuum Z.

Confluent images of the hairy arc

Abstract We introduce a class of smooth dendroids (called weak hairy arcs ) which generalizes the hairy arc, and show that the confluent images of the hairy arc are contained in this class. We show that for confluent maps of the hairy arc, the properties of being open, light, and finite-to-one are equivalent.

Jones’s Set Function \(\mathcal {T}\)

We prove basic results about the set function \(\mathcal {T}\) defined by F. Burton Jones. We define this function on compacta and then we concentrate on continua. In particular, we present some of the well known properties (such as connectedness im kleinen, local connectedness, semi-local connectedness, etc.) using the set function \(\mathcal {T}\). The notion of aposyndesis is the main motivation of Jones to define this function. We study the idempotency of T on products, cones and suspensions. We present some properties of a continuum assuming the continuity of the set function T and examples of classes of continua for which T is continuous. We give three decomposition theorems using \(\mathcal {T}\). We also present some applications.

Inverse Limits and Related Topics

We present basic results about inverse limits and related topics. An excellent treatment of inverse limits, distinct from the one given here, was written by W. Tom Ingram. First, we present some basic results of inverse limits. Then a construction and a characterization of the Cantor set are given using inverse limits. With this technique, it is shown that a compactum is a continuous image of the Cantor set. Next, we show that inverse limits commute with the operation of taking finite products, cones and hyperspaces. We give some properties of chainable continua. We study circularly chainable and P -like continua. We end the chapter presenting several properties of universal maps and AH-essential maps.

A Theorem of E. G. Effros

We present a topological proof of a Theorem by E. G. Effros and a consequence of it, due to C. L. Hagopian, which has been very useful in the theory of homogeneous continua. We present the proof of Effros’s result given by Fredric G. Ancel.

n-Fold Hyperspace Suspensions

In 1979 Sam B. Nadler, Jr. introduced the hyperspace suspension of a continuum to present examples of disk-like continua with the fixed point property.