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 Specific useful sets Booleans Integers Reals Real plane Closed interval Open interval Semi-open interval

Relations of Sets Product Binary relation Equivalence relation Reflexive Symmetric Transitive Equivalence relation

Functions Domain Codomain

Function Properties Injection Inverse function Surjection Bijection

Convexity Visibility Observation point Convex hull

Topological Spaces Topological properties Topology Point-set topology

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

Usual Topology Euclidean plane Open disk Validate T 1 and T 2

Travel Time Topology Travel time relation Neighborhoods Symmetric All time zones

Near Point X x Every neighborhood of x contains some point of 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

Properties of A Topological Space Open set Closed set Closure

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

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

Future Topics Combinatorial topology Network spaces Metric spaces Graph Metric spaces Fractal geometry

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