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MTH 304: General Topology Semester 2, 2017-2018

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MTH 304: General Topology Semester 2, 2017-2018 MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds........................... 76 V. Instructor Notes 80 2 I. Continuous Functions 1. First Definitions Following [Crossley, Section 2.1,2.2] Let S ⊂ R. A function in this section will be a real-valued function whose domain is S. 1.1. Remark: Consider two graphs (one continuous and other discontinuous at x = 1). Continuity means that we can draw the graph of f without lifting our pencil. ie. If we approach a point on the x axis from either direction, the value of f(x) should be `predicted' by the values of f(y) where y is near x. Furthermore, Continuity is a local property. 1.2. Definition: A function f : S ! R is said to be sequentially continuous at a 2 S if, for any sequence (xn) ⊂ S such that xn ! a, we have f(xn) ! f(a). 1.3. Example: f(x) = x=jxj for x 6= 0 and f(0) = 1 (i) If we choose a = 0 and xn = 1=n, then f(a) = lim f(xn) (ii) However, if we choose xn = −1=n, then f(a) 6= lim f(xn). So f is not sequentially continuous. 1.4. Definition: A function f : S ! R is said to be continuous at a if, for every > 0; 9δ > 0 such that jx − aj < δ ) jf(x) − f(a)j < (∗) 1.5. Example: (i) f(x) = x2 is continuous at 2 (a) If a = 0; = 1, we want δ > 0 such that (∗) holds. ie. We want jxj < δ ) jx2j < 1 Since jx2j = jxj2, we may choose δ = 1. (b) If a = 2; = 1, we want δ > 0 such that (∗) holds. ie. We want jx − 2j < δ ) jx2 − 22j < 1 Notice that δ = 1 does not work, because if x = 2:9 then x2 ≈ 9. However, jx2 − 22j = jx − 2jjx + 2j So 9δ > 0 that works. 3 (ii) f(x) = x2 if x 6= 0 and f(0) = 0:5 is discontinuous at 1. (a) If = 1, then δ = 0:5 works because if jxj < 0:5 ) jx2j < 0:25 < 1; and jf(0)j = 0:5 < 1 (b) However, if = 0:2, then no δ > 0 works because if jxj < δ, then we may choose small enough x so that jxj < 0:5, so that jx2j < 0:25 and hence jx2 − 0:5j > 0:25 So f is discontinuous at 0. 1.6. Theorem: f is continuous at a if and only if it is sequentially continuous at a Proof. (i) Suppose f is continuous at a and (xn) ⊂ S is a sequence such that xn ! a. WTS: f(xn) ! f(a), so choose > 0, then 9δ > 0 such that jx − aj < δ ) jf(x) − f(a)j < For this δ > 0; 9N 2 N such that jxn − aj < δ for all n ≥ N. Hence, jf(xn) − f(a)j < 8n ≥ N This is true for any > 0 so f(xn) ! f(a) (ii) Suppose f is sequentially continuous at a, but it is not continuous at a, then 9 > 0 for which no δ works. Hence, δ = 1=n does not work, so 9xn 2 S such that jxn − aj < 1=n; but jf(xn) − f(a)j ≥ Clearly, xn ! a, but f(xn) does not converge to f(a). Hence, f is not sequentially continuous - a contradiction. (End of Day 1) 2. Open Sets [Crossley, Section 2.3] 2.1. Remark: − δ definition says that f is continuous at a if and only if, for any > 0; 9δ > 0 such that x 2 (a − δ; a + δ) ) f(x) 2 (f(a) − , f(a) + ) 2.2. Definition: (i) Open interval 4 (ii) Open set: A set U ⊂ R is open if and only if it can be written as a union of open intervals. (Note: We are not restricting ourselves to finite unions. ie. We are referring to `arbitrary' unions) 2.3. Theorem: A set U ⊂ R is open iff for all x 2 U; 9δx > 0 such that (x−δx; x+δx) ⊂ U Note: The value of δx depends on x. Proof. (i) Suppose that, for any x 2 U; 9δx > 0 such that (x − δx; x + δx) ⊂ U, then [ U = (x − δx; x + δx) x2U so U is open. S (ii) Conversely, if U is open, then write U = α2J Iα, where each Iα is an open interval. If x 2 U, then 9α 2 J such that x 2 Iα. Write Iα = (a; b), then a < x < b, so δx = minfjx − aj=2; jb − xj=2g works. 2.4. Examples: (i)( a; b) (ii) A closed interval (or even a half-open interval) is not open. (iii) f0g is not open. A finite set is not open. 2.5. Proposition: (i) An arbitrary union of open sets is open. (ii) A finite intersection of open sets is open. Proof. (i) is obvious, so we prove (ii): By induction, it suffices to consider the case of two sets, U1;U2 say. WTS: U1 \ U2 is open, so fix x 2 U1 \ U2, then 9δ1; δ2 > 0 such that (x − δi; x + δi) ⊂ Ui; i = 1; 2. Then if δ = minfδ1; δ2g, then (x − δ; x + δ) ⊂ U1 \ U2, which verifies Theorem 2.3. 2.6. Example: A countable intersection of open sets may not be open. Un = (−1=n; 1=n) 2.7. Definition: A set F ⊂ R is closed if F c is open. 2.8. Examples: (i) Closed interval (ii) [2; 1) is closed. (iii) Arbitrary intersection of closed sets is closed. (iv) Finite union of closed sets is closed. (v) [1; 2) is neither open nor closed. 5 3. Continuity by Open Sets [Crossley, Section 2.4] 3.1. Definition: Let f : X ! Y be a function between two sets and A ⊂ Y , then f −1(A) = fx 2 X : f(x) 2 Ag Note: This definition does not imply that f −1 exists as a function. It is simply notation. 3.2. Example: f(x) = x2 − x = x(x − 1) (i) f −1(R) = R (ii) f −1(;) = ; (iii) f −1[−1; 1) = R (iv) f −1[0; 1) = R n (0; 1) (v) f −1(f0g) = f0; 1g 3.3. Proposition: Let f : X ! Y and fAα : α 2 Jg be a collection of subset of Y , then (i) f −1(;) = ; (ii) f −1(Y ) = X −1 T T −1 (iii) f ( Aα) = f (Aα) −1 S S −1 (iv) f ( Aα) = f (Aα) [HW] 3.4. Theorem: Let f : R ! R, then f is continuous if and only if f −1(U) is open whenever U is open. Proof. (i) Suppose f is continuous and U is open in R. WTS: f −1(U) is open, so fix x 2 f −1(U). So that f(x) 2 U, so 9 > 0 such that (f(x) − , f(x) + ) ⊂ U By definition of continuity, 9δ > 0 such that jy − xj < δ ) jf(y) − f(x)j < So if y 2 (x − δ; x + δ), then f(y) 2 (f(x) − , f(x) + ) ⊂ U. Hence, (x − δ; x + δ) ⊂ f −1(U) This is true for any x 2 f −1(U). By Theorem 2.3, f −1(U) is open. 6 (ii) Suppose f −1(U) is open whenever U is open. Fix a 2 R; > 0. Then U = (f(a) − , f(a) + ) is open in R so f −1(U) is open. Since a 2 f −1(U); 9δ > 0 such that (a − δ; a + δ) ⊂ f −1(U) Hence, if x 2 R such that jx − aj < δ, then jf(x) − f(a)j < . (End of Day 2) 7 II. Topological Spaces 1. Definition and Examples 1.1. Definition: Let X be a set. A collection τ of subsets of X is called a topology on X if (i) ;;X 2 τ (ii) If U1;U2 2 τ, then U1 \ U2 2 τ S (iii) If fUα : α 2 Jg is an arbitrary collection of sets in τ, then α2J Uα 2 τ The pair (X; τ) is called a topological space, and members of τ are called open sets in X. 1.2. Examples: (i) X = R and τ = the collection of open sets in R (as defined in the previous section) is a topological space. This is called the usual topology on R (ii) Let X = R2. (a) Fix a := (a1; a2) 2 X; r > 0. An open disc in X centered at x of radius r is the set 2 p 2 2 B(a; r) := f(x1; x2) 2 R : (x1 − a1) + (x2 − a2) < rg (b) A set U ⊂ R2 is said to be open if it is a union of open discs. As in Theorem 2.3, a set U ⊂ R2 is open if and only if, for any a 2 U; 9r > 0 such that B(a; r) ⊂ U.
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