## 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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