Local homeomorphisms

<p>This dissertation consists of (A) a history of local homeomorphisms and (B) my research pertaining thereto.</p> <p>In part (A), research concerning local homeomorphisms is traced from 1906—beginning with the work of Hadamard—up to the present. It is shown that a few major questions such as, “When is a local homeomorphism a homeomorphism?,” generated the major portion of the research activity.</p> <p>In part (B), three major concepts are defined and/or considered, namely, H-connected spaces, induced decompositions, and locally separating sets.</p> <p>We say that a T₂ connected space X is H-connected iff any proper local homeomorphism of a connected space onto X is a homeomorphism. It is shown that H-connectedness is a legitimate generalization of simple connectedness in the context of locally path connected spaces, and properties of H-connected spaces are determined. For example, the union of two “nice” H-connected spaces is H-connected provided their intersection is connected.</p> <p>It is proved that any proper local homeomorphism g of a 1° T₂ space X onto a path connected space Y induces a decomposition X₁,...,X_k of X into connected sets X_i such that ^gx_i is a 1-1 map onto Y. Generalizations and refinements of this result are obtained.</p> <p>We say that a closed subset M of a space X separates X locally iff there is an open set V ⊃ M such that V-M = A ∪ B where A and B are separated sets such that M ⊂ A ∩ B We prove that if X is any T₂ space with a subset which separates X locally but not globally, then X has a connected k-fold covering space (X*,p) for each k which decomposes into k mutually disjoint connected sets X_i such that ^Px_i is a 1-1 map onto X; moreover, the sets X_i are mutually homeomorphic.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematical Sciences

Year

1977

Author dc:creator

• Jungck, Gerald F.

Contributors dc:contributor

• Louis F. McAuley
• Ross Geoghegan
• Patricia T. McAuley

• Homeomorphisms

Repository record dc:identifier

https://orb.binghamton.edu/dissertation_and_theses/399

OAI identifier oai:identifier

oai:orb.binghamton.edu:dissertation_and_theses-1405