Thin-Type Dense Sets and Related Properties
Type of Degreedissertation
Mathematics and Statistics
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Dense sets in topological spaces may be thought of as those which are ubiquitous. We discuss dense sets in product spaces which also have a thin-type property, making them in some sense rare or spread out. Thin-type properties include the previously studied properties thin, very thin, and slim. We construct examples showing that even in a separable space, there may be no countable very thin or slim dense set. We also define and discuss the properties $<\!\kappa$-thin, codimension 1 slim, and superslim. The definition of $<\!\kappa$-thin is between those of thin and very thin; the definition of superslim is between very thin and slim. Codimension 1 slim is slightly weaker than slim, in that since only some of the cross-sections are required to be nowhere dense, it is possible for a space to have a codimension 1 slim dense set but no slim dense set. We give some results about the existence of a $<\!\kappa$-thin dense set, in one case relating this to the existence of a very thin dense set. We show that a superslim dense set in a finite power of X is related to the existence of a certain type of collection of nowhere dense subsets of X. The criteria (GC) and (NC), relating to collections of nowhere dense sets, are discussed. These were shown by Gruenhage, Natkaniec, and Piotrowski to imply the existence of a slim dense set in certain products. We consider when a space can satisfy (GC) with a collection of finite sets, and show that a collection witnessing (GC) cannot be uncountable if X is first countable and separable. We particularly consider ordered spaces, and characterize the linearly ordered and generalized ordered spaces which satisfy (GC), along with the linearly ordered spaces which satisfy (NC). The latter is connected to properties of ultrafilters. We also introduce a connection between a stronger version of (GC) and GN-separability.