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Topologies generated by relations

Published online by Cambridge University Press:  17 April 2009

Raymond E. Smithson
Affiliation:
Univėrsity of Wyoming, Wyoming, USA.
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Abstract

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Let R be a relation on a set X, and if AX set RA = {x ∣ (x, α) ∈ R for some α ∈ A} and AR = {x ∣ (α, x) ∈ R for some α ∈ A}. Also A is called an antiset in case no two distinct elements of A are related. If A is a collection of antisets, then we generate a topology T(A) by taking sets of the form RA or AR (or X or ø) as subbasic open sets. Then conditions are given under which this topology satisfies separation axioms, or is compact or connected. For example. Theorem: Let A contain the singletons. If for each xX and yX \ x, there is a zX such that (x, z) ∈ R ((z, x) ∈. R) and (y, z) ∉ R ((z, y) ∉ R), then T(A) is a T1-topology. The conditions used to obtain compactness or connectedness are analogous to the conditions used to get the same properties for the order topology on a totally ordered set. Finally, by modifying the definition of T(A) slightly, we obtain conditions so that if X is a tree and R the cutpoint order, then T(A) is the original topology.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

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