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## Orderable topological spaces

Let (X , ਹ) be a topological space. If < is a total ordering on X , then (X , ਹ, <) is said to be an ordered topological space if a subbasis for ਹ is the collection of all sets of the form {x ∊ x | x < t} or [x ∊ x | t < x} where t ∊ X . The pair (X , ਹ) is said to be an orderable topological space if there exists a total ordering, < , on X such that (X , ਹ, <) is an ordered topological space.

Definition: Let T be a subspace of the real line ǀR . Let Q be the union of all non-trivial components of T , both of whose end points belong to C1ıʀ(C1ıʀ(T) -T).

The following characterization of orderable sub-spaces of ǀR is due to M. E. Rudin.

Theorem: Let T be a subspace of ǀR with the relativized usual topology. Then T is orderable if and only if T satisfies the following two conditions:

(1) If T - Q is compact and (T-Q) ก Clıʀ(Q) = Ø then either Q = Ø or T - Q = Ø

(2) If I is an open interval of ıʀ and p is an end point of I and if {p} U(I ก(T-Q)) is compact and {p} =Clıʀ(IกQ)ก C1ıʀ(I ก(T-Q)), then p ∉ T or {p} is a component of T.

This theorem enables us to prove a conjecture of I.L. Lynn, namely Corollary: if T contains no open compact sets then T is totally orderable.

If T is a subspace of an arbitrary ordered topological space a generalization of the theorem can be made. The generalized theorem is stated and some examples are given. / Science, Faculty of / Mathematics, Department of / Graduate

Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/34573 |

Date | January 1971 |

Creators | Galik , Frank John |

Publisher | University of British Columbia |

Source Sets | University of British Columbia |

Language | English |

Detected Language | English |

Type | Text, Thesis/Dissertation |

Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |

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