General Topology

Start reading → Topological Structures

1. Front matter
   1. Introduction
   2. Advice to the reader
   3. Advice to the reader
2. ITopological Structures
   1. Topological Structures
   2. § 1. Open sets, neighbourhoods, closed sets24
   3. § 2. Continuous functions30
   4. § 3. Subspaces, quotient spaces24
   5. § 4. Product of topological spaces21
   6. § 5. Open mappings and closed mappings24
   7. § 6. Filters30
   8. § 7. Limits23
   9. § 8. Hausdorff spaces and regular spaces30
   10. § 9. Compact spaces and locally compact spaces52
   11. § 10. Proper mappings30
   12. § 11. Connectedness29
   13. Historical Note
3. IIUniform Structures
   1. Uniform Structures
   2. § 1. Uniform spaces14
   3. § 2. Uniformly continuous functions27
   4. § 3. Complete spaces48
   5. § 4. Relations between uniform spaces and compact spaces24
   6. Historical Note
4. IIITopological Groups
   1. Topological Groups
   2. § 1. Topologies on groups11
   3. § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups46
   4. § 3. Uniform structures on groups18
   5. § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators33
   6. § 5. Infinite sums in commutative groups19
   7. § 6. Topological groups with operators; topological rings division rings and fields24
   8. § 7. Inverse limits of topological groups and rings15
   9. Historical Note
5. IVReal Numbers
   1. Real Numbers
   2. § 1. Definition of real numbers10
   3. § 2. Fundamental topological properties of the real line15
   4. § 3. The field of real numbers2
   5. § 4. The extended real line11
   6. § 5. Real-valued functions31
   7. § 6. Continuous and semi-continuous real-valued functions15
   8. § 7. Infinite sums and products of real numbers20
   9. § 8. Usual expansions of real numbers; the power of R3
   10. Historical Note
6. VOne-parameter groups
   1. One-parameter groups
   2. § 1. Subgroups and quotient groups of R13
   3. § 2. Measurement of magnitudes5
   4. § 3. Topological characterization of the groups R and T5
   5. § 4. Exponentials and logarithms1
   6. Historical Note
7. VIReal number spaces and projective spaces
   1. Real number spaces and projective spaces
   2. § 1. Real number space R^n11
   3. § 2. Euclidean distance, balls and spheres12
   4. § 3. Real projective spaces14
   5. Historical Note
8. VIIThe additive groups $\mathbf{R}^n$
   1. The additive groups $\mathbf{R}^n$
   2. § 1. Subgroups and quotient groups of $\mathbf{R}^n$19
   3. § 2. Continuous homomorphisms of $\mathbf{R}^n$ and its quotient groups10
   4. § 3. Infinite sums in the groups $\mathbf{R}^n$9
   5. Historical Note
9. VIIIComplex numbers
   1. Complex numbers
   2. § 1. Complex numbers, quaternions9
   3. § 2. Angular measure, trigonometric functions5
   4. § 3. Infinite sums and products of complex numbers9
   5. § 4. Complex number spaces and projective spaces3
   6. Historical Note
10. IXUse of real numbers in general topology
    1. Use of real numbers in general topology
    2. § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces19
    3. § 2. Metric spaces and metrizable spaces33
    4. § 3. Metrizable groups, valued fields, normed spaces and algebras40
    5. § 4. Normal spaces25
    6. § 5. Baire spaces16
    7. § 6. Polish spaces, Souslin spaces, Borel sets48
    8. App. 0. Infinite products in normed algebras16
    9. Historical Note
11. XFunction spaces
    1. Function spaces
    2. § 1. The uniformity of $\mathscr{G}$-convergence33
    3. § 2. Equicontinuous sets24
    4. § 3. Special function spaces31
    5. § 4. Approximation of continuous real-valued functions18
    6. Historical Note
12. Indexes
    1. Index of notation
    2. Index of notation
    3. Index of terminology
    4. Index of terminology