arXiv subjects

Scott Varagona

Publications and source records attributed to Scott Varagona.

3 recordsLinked to original sources

arXiv2018

Mahavier Products, Idempotent Relations, and Condition $\Gamma$

Clearly, a generalized inverse limit of metrizable spaces indexed by $\mathbb N$ is metrizable, as it is a subspace of a countable product of metrizable spaces. The authors previously showed that all idempotent, upper semi-continuous, surjective, continuum-valued bonding functions on $[0,1]$ (besides the identity) satisfy a certain Condition $\Gamma$; it follows that only in trivial cases can a generalized inverse limit of copies of ([0,1]) indexed by an uncountable ordinal be metrizable. The authors show that Condition $\Gamma$ is in fact guaranteed by much weaker criteria, proving a more general metrizability theorem for certain Mahavier Products.

math.GN↗

arXiv2016

Idempotent bonding relations are nontrivial if and only if they satisfy condition Gamma

A relation $f\subseteq X^2$ satisfies condition $Γ$ if there exist distinct $x,y\in X$ with $\langle x,x\rangle ,\langle x,y\rangle ,\langle y,y\rangle \in f$. The authors improve a previous result by characterizing nontrivial idempotent bonding relations on compact Hausdorff spaces as those satisfying condition $Γ$.

math.GN↗

arXiv2015

Generalized Inverse Limits Indexed by Totally Ordered Sets

Although inverse limits with factor spaces indexed by the positive integers are most commonly studied, Ingram and Mahavier have defined inverse limits with set-valued functions broadly enough for any directed index set to be used. In this paper, we investigate generalized inverse limits whose factor spaces are indexed by totally ordered sets. Using information about the projections of such inverse limits onto finitely many coordinates, we generalize various well-known theorems on connectedness in inverse limits. Moreover, numerous theorems and examples are given addressing the special case of an inverse limit with a single idempotent surjective u.s.c. bonding function.

math.GN↗