Physical Sciences and Mathematics Commons^™

A Novel Proof Of The Heine-Borel Theorem, Matthew Macauley, Brian Rabern, Landon Rabern

Publications

Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.

Equivalences On Acyclic Orientations, Matthew Macauley, Henning S. Mortveit

Publications

The cyclic and dihedral groups can be made to act on the set Acyc(Y ) of acyclic orientations of an undirected graph Y , and this gives rise to the equivalence relations ∼κ and ∼δ, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers.

In this paper we construct the graphs C(Y ) and D(Y ) with vertex sets Acyc(Y ) and whose connected components encode the equivalence classes. The number of connected components …