Answer: 1. Topology of Metric Spaces 1 2. Topological Spaces 3 3. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. Infinitude of Prime Numbers 6 5. Product Topology 6 6. Subspace Topology 7 7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9 8. Continuous Functions 12 8.1. A Theorem of Volterra Vito 15 9. Homeomorphisms 16 10. Product, Box, and Uniform Topologies 18 11. Compact Spaces 21 12. Quotient Topology 23 13. Connected and Path-connected Spaces 27 14. Compactness Revisited 30 15. Countability Axioms 31 16. Separation Axioms 33 17. Tychonoff’s Theorem 36 References 37 1. Topology of Metric Spaces A function d : X × X → R+ is a metric if for any x, y, z ∈ X, (1) d(x, y) = 0 iff x = y. (2) d(x, y) = d(y, x). (3) d(x, y) ≤ d(x, z) + d(z, y). We refer to (X, d) as a metric space. Exercise 1.1 : Give five of your favourite metrics on R 2 . Exercise 1.2 : Show that C[0, 1] is a metric space with metric d∞(f, g) := Reply
Answer:
1. Topology of Metric Spaces 1
2. Topological Spaces 3
3. Basis for a Topology 4
4. Topology Generated by a Basis 4
4.1. Infinitude of Prime Numbers 6
5. Product Topology 6
6. Subspace Topology 7
7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9
8. Continuous Functions 12
8.1. A Theorem of Volterra Vito 15
9. Homeomorphisms 16
10. Product, Box, and Uniform Topologies 18
11. Compact Spaces 21
12. Quotient Topology 23
13. Connected and Path-connected Spaces 27
14. Compactness Revisited 30
15. Countability Axioms 31
16. Separation Axioms 33
17. Tychonoff’s Theorem 36
References 37
1. Topology of Metric Spaces
A function d : X × X → R+ is a metric if for any x, y, z ∈ X,
(1) d(x, y) = 0 iff x = y.
(2) d(x, y) = d(y, x).
(3) d(x, y) ≤ d(x, z) + d(z, y).
We refer to (X, d) as a metric space.
Exercise 1.1 : Give five of your favourite metrics on R
2
.
Exercise 1.2 : Show that C[0, 1] is a metric space with metric d∞(f, g) :=