Principles of GIS
Fundamental spatial concepts – Part II
Shaowen Wang
CyberInfrastructure and Geospatial Information Laboratory (CIGI)
Department of Geography and Geographic Information Science
Department of Computer Science
Department of Urban and Regional Planning
National Center for Supercomputing Applications (NCSA)
University of Illinois at Urbana-Champaign
October 8, 2013
Types of Sets
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Specific useful sets
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Booleans
Integers
Reals
Real plane
Closed interval
Open interval
Semi-open interval
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Relations of Sets
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Product
Binary relation
– Reflexive
– Symmetric
– Transitive
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Equivalence relation
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Functions
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Domain
Codomain
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Function Properties
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Injection
– Inverse function
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Surjection
Bijection
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Convexity
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Visibility
Observation point
Convex hull
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Topological Spaces
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Topological properties
Topology
Point-set topology
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Neighborhood

Neighborhoods
– A collection of subsets of a given set of
points S
T1: Every point in S is in some neighbor
 T2: The intersection of any two neighborhoods
of any point x in S contains a neighborhood of
x
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Usual Topology
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Euclidean plane
Open disk
Validate T 1 and T 2
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Travel Time Topology

Travel time relation
– Symmetric
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Neighborhoods
– All time zones
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Near Point

X
– Subset of points in a topological space

x
– An individual point in the topological space

Every neighborhood of x contains some
point of X
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Properties of A
Topological Space
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Open set
Closed set
Closure
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Properties of A
Topological Space

Open set
– Every point of a set can be surrounded by a
neighborhood that is entirely within the set

Closed set
– A set contains all its near points

Closure (X -)
– The union of a point set with the set of all its near
points
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Properties of A Topological
Space – continued

Interior (X o) of a point set
– Consists of all points that belong to the set and
are not near points of the complement of the set

Boundary of a point set (∂X)
– Consists of all points that are near to both the set
and its complement

Connectedness
– Partition into two non-empty disjoint subsets: A
and B
– Either A contains a point near B
– Or B contains a point near A
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Future Topics
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Combinatorial topology
Network spaces
– Graph
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Metric spaces
Fractal geometry
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End of This Topic
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