Integrating Past, Present, and Future Approaches - Spatial Mathematics: Theory and Practice through Mapping - page 236

direction. In the Manhattan tour bus example, the stops are nodes and the
path of the bus indicates edges joining pairs of stops. The edges are directed,
with direction on an edge corresponding to direction of the bus. Two con-
secutive stops are represented as nodes that are said to be adjacent. The edge
linking these stops is said to be incident with each node. Graphs from graph
theory are at the heart of “discrete mathematics” (as opposed to “continuous
mathematics”) and they are central in understanding many concepts in com-
puter science as well as network analysis.
The graph-theoretic concept of connectivity was the basis for the routing
activities. That concept is not only basic in graph theory but also in general
topology. Because it is one central topic in both mathematical subfields, there
are unfortunate instances in the published literature that make reference to
“network topology” when what is meant in terms of mathematical terminol-
ogy is “network connection pattern.” Issues such as this one arise from dif-
ficulties traversing disciplinary boundaries.
As graph theory is a base for the study of discrete mathematics, general
topology (Kelley, 1955; Bourbaki, 1966) is a base for the study of continuous
mathematics. Topology studies properties preserved under continuous trans-
formations (such as, for example, a stretching deformation). It is based on ideas
that are fundamental to mapping as well as to geometry and set theory—ideas
such as space, dimension, and transformation. Topology contains a number
of subfields that one might study including point-set (general) topology, alge-
braic topology, and geometric topology. In application, one sees most often
elements of topological spaces, from point-set topology, that involve connect-
edness, or compactness, or that involve a specialized topological space called
a metric space. In this topic, we have already drawn heavily on topological
concepts beyond network connection pattern. The Jordan Curve Theorem is
a topological theorem that has found application in geography in problems
associated with street-addressing and geocoding. In our view of the Four-
Color Theorem, another theorem from topology, we found application to map
coloring and the associated communication of spatial content using color.
Indeed, the whole notion of transformation lies at the heart of both topology and
spatial science. We considered stereographic projection from the North Pole of
the sphere into a tangent plane at the South Pole of the sphere and saw that all
points on the sphere, except the North Pole, map to that plane. We noted that
the lack of mapping the projection pole into the plane was rooted in Euclidean
geometry; in the non-Euclidean world, things can be different. In the inverse ste-
reographic transformation, all points of the plane, map in a one-to-one fashion to
the spherical surface, minus the North Pole. The two surfaces are topologically
equivalent (homeomorphic). Add the missing point to the spherical surface (at the
North Pole) and what had been a non-compact surface (a plane, of infinite extent)
becomes compact. The topological concept of compactness underlies this trans-
formation. This idea is the basis of the so-called “One-point Compactification
Theorem” of topology (Alexandroff Extension, cited recently in 2011).