Book:Steven A. Gaal/Point Set Topology

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Steven A. Gaal: Point Set Topology

Published $\text {1964}$, Dover

ISBN 0-486-47222-1.

Subject Matter


Introduction to Set Theory
1. Elementary Operations on Sets
2. Set Theoretical Equivalence and Denumerbility
3. The Axiom of Choice and Its Equivalents
Chapter $\text {I}$ Topological Spaces
1. Open Sets and Closed Sets
2. Interior, Exterior, Boundary and Closure
3. Closure Operators
4. Bases and Subbases
5. Topologies on Linearly Ordered Sets
6. Metric Spaces
7. Neighborhood Filters
8. Uniform Structures
9. Simple Results on Uniform Structures and Uniform Spaces
10. Subspaces
11. Product Spaces
12. Products of Uniformizable Spaces
13. Inverse and Direct Images of Topologies
14. Quotient Spaes
Chapter $\text {II}$ Separation Properties
1. $(T_0)$ and $(T_1)$ Axioms, Hausdorff Spaces
2. $(T_3)$ Spaces, Regular and Semiregular Spaces
3. $(T_4)$ Spaces and Normal Spaces
4. Point-Finite and Star-Finite Open Coverings
5. $(T_5)$ Spaces and Completely Normal Spaces
6. Separated Sets
7. Connected Spaces and Sets
8. Maximal Connected Subsets
9. $(T)$ Axiom and Complete Regularity
10. Uniformization and Axiom $(T)$
11. Axioms of Separation in Product Spaces
12. Separable Spaces and Countability Axioms
Chapter $\text {III}$ Compactness and Uniformization
1. Compactness
2. Compact Metric Spaces
3. Subspaces and Separation Properties of Compact Spaces
4. The Product of Compact Topological Spaces
5. Locally Compact Spaces
6. Paracompactness and Full-Normality
7. The Equivalence of Paracompactness and Full-Normality
8. Metrizable Uniform Structures and Structure Gages
9. Metrizability Conditions
Chapter $\text {IV}$ Continuity
1. Functional Relations and Functions
2. Local Continuity
3. Continuous Functions
4. Homeomorphisms, Open and Closed Maps
5. Real-Valued Functions
6. Continuity and Axioms of Separation
7. Continuity and Compactness
8. Continuity and Connectedness
9. Continuity in Product Spaces
10. Uniform Continuity and Equicontinuity
11. The Topology of Uniform Convergence
12. The Algebra of Continuous Function
Chapter $\text {V}$ Theory of Convergence
1. Filters and Nets
2. Convergence of Filters, Nets and Sequences
3. Ultrafilters and Universal Nets
4. Bounds, Traces, and Products of Filters
5. Applications of Filters and Nets to Compactness
6. Cauchy Filters and Complete Spaces
7. Completion of Metric Structures
8. Baire's Category Theorem, the Principles of Uniform Boundedness and of the Condensation of Singularities
9. Completions and Compactifications
author index
subject index



Zero Element of Union and Intersection

Introduction to Set Theory: $1$. Elementary Operations on Sets:

The operations $A \cup B$ and $A \cap B$ are meaningful for every pair of sets $A, B$ ... The empty set $\O$ plays the role of the zero element.

Cartesian Product of Countable Sets is Countable: Formal Proof 2

Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability:

If $A$ and $B$ are denumerable, then so are $A \cup B$ and $A \times B$.
For instance, if $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ are enumerations of $A$ and $B$, then the map $f$ given by the rule
$\map f {\tuple {a_k, a_l} } = \dfrac {\paren {k + l - 1} \paren {k + l - 2} } 2 + \dfrac {l + \paren {-1}^{k + 1} } 2 k + \dfrac {1 + \paren {-1}^{k + l - 1} } 2 l$
gives an enumeration of $\tuple {A \times B}$.

Source work progress

From here on in there is much work to do on Axiom:Axiom of Choice.
Starting on Chapter $\text I$ with Next: