Set Function T

Abstract

Aims and ScopeThe Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High-quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.

Uniformly refinable maps

“…The next result extends [14,Corollary 4.1.26] to uniformly monotonely refinable maps. Let X and Y be compacta and let f : X → → Y be a surjective map.…”

“…The following theorem extends [14,Theorem 4.1.32] to uniformly atomically refinable maps and gives a partial positive answer to [14,Question 8.1.18]. The following theorem is an extension to closed sets of [7,Theorem 1 ].…”

“…Let X and Y be compacta and let f : X → → Y be a uniformly refinable map. Then for every subcontinuum Q of Y , there exists a subcontinuum [14,Theorem 1.6.7], by [4, 3.1.23 and 1.6.1], there exists a cofinal subset Λ…”

“…Burton Jones defined what are now known as the set functions T and K. He used these set functions to study aposyndesis on continua. Both functions have been used to study continua, mainly T [14]. These set functions are considered "dual" functions.…”

“…Then X \ L = A ∪ B, where A and B are disjoint open subsets of X. By the Boundary Bumping Theorem[14, Theorem 1.4.36], if R is a component of Cl X (A), then R ∩ L = ∅ and if S is a component of Cl X (B), then S ∩ L = ∅. Let T = {R | R is a component of Cl X (A)} and let W = {S | S is a component of Cl X (B)}.…”