# Text: Elementary Topology

### Who is the book for?

The purpose of this text is to provide an one semester introduction to topology at an advanced undergraduate, or beginning graduate level.

For some students this will be a first and last exposure to topology. If so, the student will leave with some interesting concrete examples, some interesting questions involving the plane and three dimensional space, and the impression that mathematical investigation of these matters is an interesting and useful endeavor.

For others, who will continue to learn about topology, our introduction will provide preparation for such basic texts as Munkres (Topology, a First Course), Willard (General Topology), or Sieradski (Introduction to Topology and Homotopy). For these students we provide a useful introduction that gives a background of good examples, yet minimizes duplication of future material.

For both groups of students we provide ample opportunity and guidance for building reasoning skills and writing proofs.

### How is this different from other books ?

The most common introduction to topology is through the study of general topology, either beginning with an axiomatic point of view, or perhaps with metric spaces. In any case, the approach is to present a slimmed version of general topology. The impression many students get from such a course is that topology is an abstract study of sets and subsets, many of which are not subsets of Euclidian space.

Our approach uses the setting of Euclidian space, mostly the line, the plane and 3-dimensional space. This provides an introduction, less abstract, than either of these others. This allows us to more quickly get to some interesting topics that one would not otherwise have time to consider. We focus questions such as: what are the different subsets of the plane (or space) and what should ``different'' mean?

Our text is ``example driven'' by considerations of subsets of n-space. As much as possible we attempt demonstrate the need for a tool before presenting it.

An additional benefit from this approach is that the study of Euclidian space interfaces with other mathematical areas familiar to the student. At various times we use geometry, analytic geometry, linear algebra, calculus, and complex numbers, to describe subsets and continuous functions. These connections show that topology is not isolated from things they already know about and also gives the students some familiar tools so they are not completely ``starting from scratch''.

### Features

• Focus on (Euclidian) n-space. This allows us to get more quickly into more interesting problems, rather than getting into long discussions of separation properties, bases and sub-bases, countability. We eliminate distractions, get more depth.
• Focus on examples. Concepts are developed to answer questions about some subsets of Euclidian space. The usual approach is to define concepts and present examples to illuminate the nature of the concept. In the usual approach, often several examples are given to illustrate a concept. In our approach, often several proofs are given to verify a single relationship
• Selection of Topics. Topics of general topology, not relevant to n-space, are not excluded, but are relegated to coordinated optional sections. For example, examples of non-Hausdorff spaces are presented as a consequence of generalization but not something that a beginning student be immediately concerned about.
• Focus on basic mathematical questions. Questions such as: What are the subsets of the plane? What are the possibilities of embedding one object into another?
• Some novel alternate approaches to standard problems. For example, the first proof offered in the text that the line is not homeomorphic to the plane is that there are non-equivalent embeddings of a three point set into the line, but all embeddings of a three point set into the plane are equivalent. The first proof that the closed unit interval is not homeomorphic to the circle is that the interval has the fixed point property and circle does not.
• Integration with undergraduate curriculum. We integrate the student's background in calculus, analytic geometry and linear algebra (especially the geometric side of linear algebra) in construction and analysis of examples.
• A selection of interesting advanced topics, manifolds, complexes, knots. ( The core material of the course is contained in the first 11 Chapters.) In each of these Chapters we present some mathematical basic mathematical results derived from the basic core material.
• An extra guided index. Besides the traditional index, a special index is provided to help the student to solve problems and investigate examples. It is an extensive list of topics, collected as examples, propositions.
• Computer generated graphics by the Author

Note: the first 11 Chapters form the core material. Sections marked with asterisk are optional topics from the point of view of general topology.

• BASIC TOPICS
• {1} Open and Closed Subsets
• {1.1}What is topology?
• {1.2}Some basic questions
• {1.3}Distance
• {1.4}Open and closed subsets of R^n
• {1.5}Open and closed subsets of subsets
• {1.6}Properties of open and closed subsets
• {1.7}Limit points
• *{1.8}General topology and Chapter 1
• {1.9}Problems for Chapter 1
• {2} Building Open and Closed Subsets
• {2.1}Some basic examples
• {2.2}Some basic properties of open and closed subsets
• {2.3}Linear and affine maps
• {2.4}Showing subsets open or closed, using affine maps
• {2.5}Cartesian products of open and closed subsets
• {2.6}Cones and suspensions
• *{2.7}General topology and Chapter 2
• {2.8}Problems for Chapter 2
• {3}Continuity
• {3.1}Definition of continuous function
• {3.2} Some basic constructions of continuous functions
• {3.3}Graphs of functions
• {3.4} More examples of continuous functions
• {3.5} More properties of continuous functions
• {3.6}The gluing lemma
• {3.7}Two more examples of continuous functions
• *{3.8}General topology and Chapter 3
• {3.9}Problems for Chapter 3
• {4}Homeomorphism
• {4.1} Homeomorphism and homeomorphism type
• {4.2}Refining basic questions about R^n
• {4.3}Intervals and homeomorphisms
• {4.4}The circle and the half-open interval
• {4.5}Topological properties
• *{4.6}General topology and Chapter 4
• {4.7}Problems for Chapter 4
• {5} Cantor Sets and Allied Topics
• {5.1}The standard Cantor set
• {5.2}Variations of definition of Cantor set in R^1
• {5.3}Cantor sets in the plane
• {5.4}Other infinite intersections
• {5.5}Cantor subsets in R^3
• *{5.6}General topology and Chapter 5
• {5.7}Problems for Chapter 5
• {6}Embeddings
• {6.1}Basic examples of embeddings
• {6.2}Equivalent embeddings; equivalent subsets
• {6.3} Some subsets of R^3 which do not embed in R^2
• {6.4}Stable embeddings
• {6.5}Answering one question and asking many more
• *{6.6}General topology and Chapter 6
• {6.7}Problems for Chapter 6
• {7}Connectivity
• {7.1}Basic properties of connected subsets
• {7.2}Components
• {7.3}Cut-points
• *{7.4}General topology and Chapter 7
• {7.5}Problems for Chapter 7
• {8}Path Connectedness
• {8.1}Path connectedness
• {8.2}Path components
• *{8.3}General topology and Chapter 8
• {8.4}Problems for Chapter 8
• {9} Closure and Limit Points
• {9.1}The closure operation
• {9.2}Boundary and interior of a subset
• {9.3}Dimension
• {9.4}Limits of sequences
• *{9.5}General topology and Chapter 9
• {9.6}Problems for Chapter 9
• {10} Compactness
• {10.1} Closed and bounded subsets
• {10.2} Properties and examples of compactness
• {10.3} Distance between subsets
• {10.4} Continua
• *{10.5} General Topology and Chapter 10
• {10.6} Problems for Chapter 10
• {11} Local Connectivity
• {11.1} Local connectedness
• *{11.2} General Topology and Chapter 11
• {11.3} Problems for Chapter 11
• {12} Space-filling Curves
• {12.1} An example of a space-filling curve
• *{12.2} General Topology and Chapter 12
• {12.3} Problems for Chapter 12
• {13} Manifolds
• {13.1} Some basic properties of manifolds
• {13.2} The torus
• {13.3} Some other manifolds
• {13.4} Further properties of manifolds
• {13.5} Smooth manifolds
• *{13.6} General Topology and Chapter 13
• {13.7} Problems for Chapter 13
• {14} Knots and Knottings
• {14.1} Knots and isotopy
• {14.2} Wild knots, smooth knots, PL knots
• *{14.3} General Topology and Chapter 14
• {14.4} Problems for Chapter 14
• {15} Simple Connectivity
• {15.1} Simple connectivity and homotopy
• {15.2} Retracts and deformation retracts
• {15.3} The circle is not simply connected
• {15.4} Some implications of the non-simple connectivity of S^1
• *{15.5} General Topology and Chapter 15
• {15.6} Problems for Chapter 15
• {16} Deformation Type
• {16.1} Definitions and examples of deformation type
• *{16.2} General Topology and Chapter 16
• {16.3} Problems for Chapter 16
• {17} Complexes
• {17.1} Simplicial complexes
• {17.2} Examples and properties of simplicial complexes
• *{17.3} General Topology and Chapter 17
• {17.4} Problems for Chapter 17
• {18} Higher Dimensions
• {18.1} The circle and torus in higher dimensions
• {18.2} The Klein bottle, the projective plane
• {18.3} The 3-dimensional sphere, S^3
• {18.4} Knotted surfaces in R^4
• *{18.5} General Topology and Chapter 18
• {18.6} Problems for Chapter 18
• {19} The Poincare Conjecture
• {19.1} What is S^3
• *{19.2} General Topology and Chapter 19
• APPENDICES
• {A}Sets and Logic
• {A.1}Logic
• {A.2}Sets, some basics
• {A.3}Cartesian products
• {A.4}Sets---topics needed for general topology
• {B}Numbers
• {C}Cardinality of Sets
• {C.1}Problems for Appendix C
• {D} Summary from Calculus
• {E}Strategy in Proof
• {E.1}Proofs in general
• {E.2}Specifics for topology proofs
• Bibliography
• Index of Examples, Remarks, and Propositions
• Subject Index