TITLE: References for Topology with applications in Engineering, Computer Science, Robotics
QUESTION [11 upvotes]: I am reading a book on motion planning for mobile robots, I have a really hard time with the mathematics. Some parts it is talking about the topology of the space and manifolds and compactness of the space.... I am too lost. I also have some problems with the math notations used in this book. There is an appendix which goes through topology and manifolds but it's only 3 pages and it really fast.
I need a very basic book or paper about topology, which I can read in about a month. Something very basic and with some real applications.
Any other comments are highly appreciated.
Edit:
I think I will need to have a basic idea about:
-  Topology
-  Manifolds
-  Group theory
-  Homology    
I have found some basic books on topology. Would you take a look at them and let me know which one would be more appropriate for me?
1- Topology for Computing
2- Computational Topology
3- Elementary Applied Topology
4- Topology and Robotics (Contemporary Mathematics), I think this one would be great if I have some basic idea about topology

REPLY [4 votes]: There's a really good book on topology in engineering mentioned by the others too:
Introduction to Topology - Pure and Applied, by Colin Adams and Robert Franzosa
The nice thing about this textbook is that it gives a detailed application to the concepts after every chapter:

Introduction
Topological Basis -> Mutation of DNA
Closure, Interior and Boundary -> Data in GIS
Creation of Spaces -> Configuration and Phase Space
Continuity -> Robotics
Metric Spaces
Connectedness -> Automated Guided Vehicles
Compactness
Dynamical Systems -> Chaos in Population Model
Homotopy -> Heartbeat Model
Fixed Point Theorem -> Economics
Embeddings -> Digital Image Processing
Knots -> Chirality in Biochemistry
Graphs -> Boiling Point of Molecules, Electronic Circuits
Manifolds -> Geometry of the Universe

For a more detailed table of contents, see http://math.umaine.edu/~franzosa/TOC.htm.