Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

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16 Topology (2301631) Definition 1.55. Let X be a space and A ⊆ X. We say that A is dense (in X) if A = X. On the other hand, A is nowhere dense if X − A is dense in X. Example 1.56. In R, Q is dense while Z is nowhere dense. Definition 1.57. Subsets A and B of a space X are said to be separated if A ∩ B = A ∩ B = ∅. Example 1.58. In R, the intervals (0, 1) and [1, 2) are disjoint but not separated, while (0, 1) and (1, 2) are separated.
Phichet Chaoha 17 Let X and Y be (nonempty) spaces. 4. Continuous Maps Definition 1.59. A map f : X → Y is said to be continuous at x ∈ X if, for any neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V . We simply say that f is continuous if it is continuous at each x ∈ X. Example 1.60. For a selfmap of R n (with the usual topology), the definition of continuity above is exactly the ε − δ definition in analysis. Theorem 1.61. A map f : X → Y is continuous iff f −1 (V ) is open in X for each open subset V of Y . Proof. (⇒) Let V ⊆ Y be an open set and x ∈ f −1 (V ). Since f is continuous at x and V is a neighborhood of f(x), there is a neighborhood U of x such that f(U) ⊆ V . Hence, U ⊆ f −1 (f(U)) ⊆ f −1 (V ) which implies that f −1 (V ) is open in X. (⇐) Let x ∈ X and V a neighborhood of f(x). It follows from the assumption that f −1 (V ) is a neighborhood of x. So, by letting U = f −1 (V ), we clearly have f(U) = f(f −1 (V )) ⊆ V and this proves the continuity of f. □
Page 1: Lecture Notes Topology (2301631) Ph
Page 5 and 6: Preliminaries Sets and Maps We assu
Page 7 and 8: CHAPTER 1 Basic Concepts in Topolog
Page 9 and 10: Phichet Chaoha 9 Example 1.12. The
Page 11 and 12: Phichet Chaoha 11 2. Basis for a To
Page 13 and 14: Phichet Chaoha 13 Exercise 1.34. Fo
Page 15: Phichet Chaoha 15 { 1 n Exercise 1.
Page 19 and 20: Phichet Chaoha 19 Exercise 1.70. Le
Page 21 and 22: Phichet Chaoha 21 5. Sequences and
Page 23 and 24: Phichet Chaoha 23 6. Product Topolo
Page 25 and 26: Phichet Chaoha 25 Moreover, if P is
Page 27 and 28: Phichet Chaoha 27 (1) The 1-sphere
Page 29 and 30: Phichet Chaoha 29 8. Metric Topolog
Page 31: Phichet Chaoha 31 where d(a, b) = m
Page 34 and 35: 34 Topology (2301631) let {X n } n
Page 36 and 37: 36 Topology (2301631) Proof. Suppos
Page 38 and 39: 38 Topology (2301631) for all x ∈
Page 41 and 42: CHAPTER 3 Connectedness Definition
Page 43 and 44: Phichet Chaoha 43 Proof. First, we
Page 45: Phichet Chaoha 45 Definition 3.35.
Page 48 and 49: 48 Topology (2301631) compact ⇓ s
Page 50 and 51: 50 Topology (2301631) If {d(a, b) |
Page 52 and 53: 52 Topology (2301631) Example 4.35.
Page 54 and 55: 54 Topology (2301631) then ̂X −
Page 56 and 57: 56 Topology (2301631) Exercise 4.59
Page 58 and 59: 58 Topology (2301631) Proof. Let (
Page 60 and 61: 60 Topology (2301631) Proof. Let (x
Page 62 and 63: 62 Topology (2301631) Exercise 5.18
Page 65 and 66: CHAPTER 6 Complete Metric Spaces De
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Phichet Chaoha 67 Example 6.15. R
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Let ϵ > 0 and N ∈ N be such that
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Phichet Chaoha 71 subset Y of R n b
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