1 research outputs found
Coproducts of proximity spaces
In this paper, we introduce coproducts of proximity spaces. After exploring
several of their basic properties, we show that given a collection of proximity
spaces, the coproduct of their Smirnov compactifications proximally and densely
embeds in the Smirnov compactification of the coproduct of the original
proximity spaces. We also show that the dense proximity embedding is a
proximity isomorphism if and only if the index set is finite. After
constructing a number of examples of coproducts and their Smirnov
compactifications, we explore several properties of the Smirnov
compactification of the coproduct, including its metrizability, connectedness
of the boundary, dimension, and its relation to the Stone-Cech
compactification. In particular, we show that the Smirnov compactification of
the infinite coproduct is never metrizable and that its boundary is highly
disconnected. We also show that the proximity dimension of the Smirnov
compactification of the coproduct equals the supremum of the covering
dimensions of the individual Smirnov compactifications and that the Smirnov
compactification of the coproduct is homeomorphic to the Stone-Cech
compactification if and only if each individual proximity space is equipped
with the Stone-Cech proximity. We finish with an example of a coproduct with
the covering dimension but the proximity dimension Comment: 24 page