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Topology:
An Inquiry-Based Approach
Steven Schlicker
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Front Matter
Preface
I
Sets and Functions
1
Sets
Introduction
The Basic Idea of Topology
Intervals
Unions, Intersections, and Complements of Sets
Cartesian Products of Sets
Summary
Exercises
2
Functions
Introduction
Composites of Functions
Inverse Functions
Functions and Sets
The Cardinality of a Set
Summary
Exercises
II
Metric Spaces
3
Metric Spaces
Introduction
Metric Spaces
The Euclidean Metric on
\(\R^n\)
Summary
Exercises
4
Applications of Metric Spaces
Introduction
The Hamming Metric
The Levenshtein Metric
5
The Greatest Lower Bound
Introduction
The Distance from a Point to a Set
Summary
Exercises
6
Continuous Functions in Metric Spaces
Introduction
Continuous Functions Between Metric Spaces
Composites of Continuous Functions
Summary
Exercises
7
Open Balls and Neighborhoods in Metric Spaces
Introduction
Neighborhoods
Continuity and Neighborhoods
Summary
Exercises
8
Open Sets in Metric Spaces
Introduction
Open Sets
Unions and Intersections of Open Sets
Continuity and Open Sets
The Interior of a Set
Summary
Exercises
9
Sequences in Metric Spaces
Introduction
Sequences and Continuity in Metric Spaces
Summary
Exercises
10
Closed Sets in Metric Spaces
Introduction
Closed Sets in Metric Spaces
Continuity and Closed Sets
Limit Points, Boundary Points, Isolated Points, and Sequences
Limit Points and Closed Sets
The Closure of a Set
Closed Sets and Limits of Sequences
Summary
Exercises
11
Subspaces and Products of Metric Spaces
Introduction
Open and Closed Sets in Subspaces
Products of Metric Spaces
Summary
Exercises
III
Topological Spaces
12
Topological Spaces
Introduction
Examples of Topologies
Bases for Topologies
Metric Spaces and Topological Spaces
Neighborhoods in Topological Spaces
The Interior of a Set in a Topological Space
Summary
Exercises
13
Closed Sets in Topological Spaces
Introduction
Unions and Intersections of Closed Sets
Limit Points and Sequences in Topological Spaces
Closure in Topological Spaces
The Boundary of a Set
Separation Axioms
Summary
Exercises
14
Continuity and Homeomorphisms
Introduction
Metric Equivalence
Topological Equivalence
Relations
Topological Invariants
Summary
Exercises
15
Subspaces
Introduction
The Subspace Topology
Bases for Subspaces
Open Intervals and
\(\R\)
Summary
Exercises
16
Quotient Spaces
Introduction
The Quotient Topology
Quotient Spaces
Identifying Quotient Spaces
Summary
Exercises
17
Compact Spaces
Introduction
Compactness and Continuity
Compact Subsets of
\(\R^n\)
An Application of Compactness
Summary
Exercises
An Application of Compactness: Fractals
18
Connected Spaces
Introduction
Connected Sets
Connected Subsets of
\(\R\)
Components
Cut Sets
The Intermediate Value Theorem and a Fixed Point Theorem
Summary
Exercises
19
Path Connected Spaces
Introduction
Path Connectedness
Path Connectedness as an Equivalence Relation
Path Connectedness and Connectedness
Path Connectedness and Connectedness in Finite Topological Spaces
Path Connectedness and Connectedness in Infinite Topological Spaces
Summary
Exercises
20
Products of Topological Spaces
Introduction
The Topology on a Product of Topological Spaces
Three Examples
Projections and Continuous Functions on Products
Properties of Products of Topological Spaces
Summary
Exercises
Applications of Products of Topological Spaces
Back Matter
Index
Section
Introduction
In this section we explore two applications of metric spaces.