Some Structure Theorems for Inverse Limits with Set-valued Functions

نویسنده

  • M. M. MARSH
چکیده

We investigate inverse limits with set-valued bonding functions. We generalize theorems of W. T. Ingram and William S. Mahavier, and of Van Nall, on the connectedness of the inverse limit space. We establish a fixed point theorem and show that under certain conditions, inverse limits with set-valued bonding functions can be realized as ordinary inverse limits. We also obtain some results that are useful in determining the existence of certain subcompacta of the inverse limit on a single space with a single set-valued bonding function. All spaces considered in this paper will be metric. A continuum is a compact, connected metric space. A continuous function f : X → Y will be referred to as a mapping. We wish to consider inverse limits on inverse sequences X1, X2, . . . of compacta with upper semi-continuous bonding functions G n : Xn+1 → 2n . These inverse limits have been called generalized inverse limits and inverse limits with set-valued bonding functions. The literature on and interest in these inverse limits is growing fairly rapidly (see [3], [4], [5], [6], [7], [8], [10], [11], [12]). Perusing these papers, one notices that there are some commonly-used notations and terminology for important concepts related to a function G : X → 2 that are defined, in a natural way, relative to X, Y , and X × Y . Ordinarily, the graph of G would lie in X × 2 with the product topology induced from the topology of X and the topology of 2 . However, we wish to view the graph of G as a subset of X × Y , and for x ∈ X, we view G(x) as a subset of Y rather than a point in 2 . For essentially all of the properties we are interested in, the topologies of X, Y , and X × Y will influence 2010 Mathematics Subject Classification. Primary 54C60, 54D80; Secondary 54B10, 54C15, 54F15, 54H25.

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تاریخ انتشار 2013