1. 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

2. Types of Sets Specific useful sets Booleans Integers Reals Real plane
Closed interval
Open interval
Semi-open interval

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

4. Functions
Domain
Codomain

5. Function Properties
Injection
Inverse function
Surjection
Bijection

6. Convexity
Visibility
Observation point
Convex hull

7. Topological Spaces
Topological properties
Topology
Point-set topology

8. 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

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

10. Travel Time Topology Travel time relation Neighborhoods Symmetric
All time zones

11. 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

12. Properties of A Topological Space
Open set
Closed set
Closure

13. 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

14. 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

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

16. End of This Topic