Date of Award

5-1968

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

L. D. Loveland

Abstract

Borsuk [ 3 ] has given interesting conditions under which a certain function space is separable (see Theorem 3. 1 ). We give a proof for Borsuk's Theorem here and we show how it can be used to establish a useful theorem on homeomorphic convergence. We illustrate the utility of the theorem on homeomorphic convergence by stating and proving several of its consequences.

For example we show that the plane (E2) does not contain uncount­ably many pairwise disjoint contina each of which contains a simple triod (Corollary 4. 1 ). We prove that in an uncountable collection G of pairwise disjoint simple closed curves in E2 "almost all" elements of G must be converged to homeomorphically "from both sides" by sequences of elements of G (see Theorem 4. 3 ). The same technique allows us to prove the nonexistence of uncountably many pairwise dis -joint wild 2 -spheres in E3.

Another interesting consequence of Borsuk's Theorem is Theorem 3. 4 which shows that in each set G consisting of uncountably many com­pact subsets of a metric space, some element of G is an element of convergence. Proofs for this theorem do not often appear in the literature, and, as far as the author knows, the proof given here does not appear in the literature.

We wish to emphasize that all the proofs given in this report were constructed by the author without reference to the literature, in fact the author was unaware of the references until after the proofs were given. We given reference at the end of the paper where proofs in the literature can be compared with the proofs given here.

We wish to emphasize that all the proofs given in this report were constructed by the author without reference to the literature, in fact the author was unaware of the references until after the proofs were given. We given reference at the end of the paper where proofs in the literature can be compared with the proofs given here.

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