Book:W.A. Sutherland/Introduction to Metric and Topological Spaces

Subject Matter

 * Metric Spaces
 * Topology

Contents

 * Introduction


 * Notation and Terminology


 * $1$. Review of some Real Analysis
 * $1.1$. Real numbers
 * $1.2$. Real sequences
 * $1.3$. Limits of functions
 * $1.4$. Continuity


 * $2$. Continuity Generalized: Metric Spaces
 * $2.1$. Motivation
 * $2.2$. Examples
 * $2.3$. Open sets in metric spaces
 * $2.4$. Equivalent metrics
 * $2.5$. Continuity


 * $3$. Continuity Generalized: Topological Spaces
 * $3.1$. Topological spaces
 * $3.2$. Bases
 * $3.3$. Sub-bases and weak topologies
 * $3.4$. Subspaces
 * $3.5$. Products
 * $3.6$. Homeomorphisms
 * $3.7$. Definitions
 * $3.8$. Quotient spaces


 * $4$. The Hausdorff Condition
 * $4.1$. Motivation
 * $4.2$. Separation axioms


 * $5$. Compact Spaces
 * $5.1$. Motivation
 * $5.2$. Definition of compactness
 * $5.3$. Compactness of $\sqbrk {a, b}$
 * $5.4$. Properties of compact spaces
 * $5.5$. Continuous maps on compact spaces
 * $5.6$. Compactness and constructions
 * $5.7$. Compact subspaces of $\R^n$
 * $5.8$. Compactness and uniform continuity
 * $5.9$. An inverse function theorem


 * $6$. Connected Spaces
 * $6.1$. Introduction
 * $6.2$. Connectedness
 * $6.3$. Path-connectedness
 * $6.4$. Comparison of definitions
 * $6.5$. Components


 * $7$. Compactness Again: Convergence in Metric Spaces
 * $7.1$. Introduction
 * $7.2$. Sequential compactness


 * $8$. Uniform Convergence
 * $8.1$. Introduction
 * $8.2$. Definition and examples
 * $8.3$. Cauchy's criterion
 * $8.4$. Uniform limits of sequences
 * $8.5$. Generalizations


 * $9$. Complete Metric Spaces
 * $9.1$. Introduction
 * $9.2$. Definition and examples
 * $9.3$. Fixed point theorems
 * $9.4$. The contraction mapping theorem
 * $9.5$. Cantor's and Baire's theorems


 * $10$. Criteria for Compactness in Metric Spaces
 * $10.1$. A general criterion
 * $10.2$. Arzelà-Ascoli Theorem


 * $11$. Appendix
 * $11.1$. Real numbers
 * $11.2$. Completion of metric spaces


 * $12$. Guide to Exercises


 * Bibliography


 * Index



Example of Preimage of Subset under Mapping: $\map g {x, y} = \tuple {x^2 + y^2, x y}$
$2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.16 \ \text {(b)}$:

Letter $\mathsf L$ and Letter $\mathsf T$ are not Homeomorphic
$3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Examples $3.6.2 \ \text{(d)}$:

Source work progress
* : up to $8.2.3$: Definition:Uniform Convergence/Real Numbers -- may be gaps
 * Redoing from start:
 * Chapter $3$ exercises to be completed:


 * : $3$: Continuity generalized: topological spaces: Exercise $3.9: 30 \ \text {(ii)}$
 * Chapter $4$ exercises to be completed:


 * : $4$: The Hausdorff condition: Exercise $4.3: 2$
 * Chapter $5$ in progress -- examples and illustrations to be completed:


 * : $5$: Compact spaces: $5.1$: Motivation: Example $5.1.2$