On free spaces

A continuum is a connected, compact, metric space. A continuum is decomposable if it is a union of two proper subcontinua. A continuum is indecomposable if it is not decomposable. A continuum is hereditarily indecomposable if each of its subcontinua is indecomposable. A space is called a Bing space if each of its components is a hereditarily indecomposable continuum. A map is a continuous function. A map is called a Bing map if each of its fibers is a Bing space.

In 1958, Brown constructed a Bing map from Rⁿ − {0} to R. In 1996, Levin proved that the set of Bing maps is a dense G_δ-subset of C(X, I) (or C(X, R)) for any compactum X. Krasinkiewicz proved the same result for the case of n-dimensional manifolds M (n ≤ 1). He introduced the “free space” to represent such a space: A space Y is called a free space if the set of Bing maps is a dense G_δ-subset of C(X, Y) for any compactum. X where C( X, Y) is the space of maps of X to Y with the uniform Pietric.

This thesis will search for more free spaces and verify that some spaces are not free spaces. These results will give a quite complete answer to the problems raised by Krasinkiewicz.

We introduce the definition of locally free space and prove that a locally free ANR space is free.

We prove that every locally finite polyhedron is a free space. By using this result, we show that several classes of spaces are free spaces: smooth dendroids, fans, 1-dimensional Peano continua and Menger cubes (in particular, the Sierpiński Curve and the Menger curve). We also prove that the hyperspaces of some spaces are free. A special bundle space and solenoids are considered.

Finally we investigate some examples of free spaces as well as examples of not free spaces. These examples provide us some insight into the extent of the class of free spaces.