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Those of us who teach Mathematics for Liberal Arts (MLA) courses often underestimate the mathematical abilities of the students enrolled in our courses. Despite the fact that many of these students suffer from math anxiety and will admit to hating mathematics, when we give them space to explore mathematics and bring their existing knowledge to the problem, they can make some amazing mathematical discoveries. Inquiry-based learning (IBL) is perfect structure to provide these type of opportunities. In this paper, we will examine one inquiry-based investigation in which MLA students were given the space to look for patterns which resulted in some original discoveries.

Abstract. Let X be a proper Busemann space. Then there is a well defined boundary, ∂X, for X. Moreover, if X is (Gromov) hyperbolic (resp. nonpositively curved), then this boundary is homeomorphic to the hyperbolic (resp. non-positively curved) boundary. §0. IntroductionThe boundary of a (Gromov) hyperbolic space (and hence of a (Gromov) hyperbolic group) was introduced in Gromov's now famous article on hyperbolic groups [G1]. Since then, this notion has received much attention and provided many interesting results (see [F] , [G1], [GH], [Sw]). This notion of boundary has been generalized to non-positively curved spaces and automatic groups (see [G2] and [NS]respectively) although it appears that the proof of this for non-positively curved spaces has not been published anywhere. However, the notion of boundary can also be extended to more general class of spaces called Busemann spaces which were defined in [Bo]. In this paper, we provide an elementary proof that the boundary of a Busemann space is well defined.Definition. Let X be a proper Busemann space, and let x 0 ∈ X. We define the boundary of X relative to x 0 as ∂ x0 X = {f : [0, ∞) → X | where f (0) = x 0 and f is an isometry} and we give ∂ x0 X the compact-open topology. Our main theorem is the following:Main Theorem. Let X be a proper Busemann space and let x 0 and x 1 be two distinct points in X. Then ∂ x0 X is homeomorphic to ∂ x1 X. (resp. non-positively curved), then ∂ x0 X is homeomorphic to the hyperbolic (resp. non-positively curved) boundary. Corollary. If a Busemann space X is (Gromov) hyperbolicIn §1 we give the basic definitions and some background material, and in §2 we define the boundary of a Busemann space. The author would like to thank his advisor, Edward C. Turner, for all his helpful suggestions and support.

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