Given a finite lexicon $L$ of relational symbols and equality, one may view the collection of all $L$-structures on the set of natural numbers $\omega$ as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on $\omega$. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on $\omega$ is ubiquitous in the set of linear orderings on $\omega$.
@article{1183743399,
author = {Bankston, Paul and Ruitenburg, Wim},
title = {Notions of Relative Ubiquity for Invariant Sets of Relational Structures},
journal = {J. Symbolic Logic},
volume = {55},
number = {1},
year = {1990},
pages = { 948-986},
language = {en},
url = {http://dml.mathdoc.fr/item/1183743399}
}
Bankston, Paul; Ruitenburg, Wim. Notions of Relative Ubiquity for Invariant Sets of Relational Structures. J. Symbolic Logic, Tome 55 (1990) no. 1, pp. 948-986. http://gdmltest.u-ga.fr/item/1183743399/