Topology + Foundations
32nd Summer Conference on Topology and Its Applications
A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C*(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an intermediate ring if C*(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P describes a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y) both satisfy P, then X satisfies T if and only if Y satisfies T. An example of a topological property being described by an algebraic property among a class of intermediate rings is that of a P-space, a Tychonoff space in which every zero-set is open. We see that the property that every prime ideal of the ring is maximal describes P-spaces among rings C(X), however for the same algebraic property does not describe P-spaces among all intermediate rings. Another example of a topological property is that of an F-space, a Tychonoff space in which disjoint co-zero sets are completely separated. We see that the property that the set of prime ideals contained in a maximal ideal form a chain describes F-spaces among all intermediate rings. We investigate what other algebraic properties describe topological properties as well as other types of relationships between algebraic properties and topological properties, and we prove some theorems about how certain topological properties relate to algebraic properties of intermediate rings.
Copyright © 2017, the Author
Sack, Joshua, "Relationships between Topological Properties of X and Algebraic Properties of Intermediate Rings A(X)" (2017). Summer Conference on Topology and Its Applications. 34.