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ELEMENTARY TOPOLOGY I 1. Introduction 3 2. Metric Spaces 4 2.1 ELEMENTARY TOPOLOGY I ALEX GONZALEZ 1. Introduction3 2. Metric spaces4 2.1. Continuous functions8 2.2. Limits 10 2.3. Open subsets and closed subsets 12 2.4. Continuity, convergence, and open/closed subsets 15 3. Topological spaces 18 3.1. Interior, closure and boundary 20 3.2. Continuous functions 23 3.3. Basis of a topology 24 3.4. The subspace topology 26 3.5. The product topology 27 3.6. The quotient topology 31 3.7. Other constructions 33 4. Connectedness 35 4.1. Path-connected spaces 40 4.2. Connected components 41 5. Compactness 43 5.1. Compact subspaces of Rn 46 5.2. Nets and Tychonoff’s Theorem 48 6. Countability and separation axioms 52 6.1. The countability axioms 52 6.2. The separation axioms 54 7. Urysohn metrization theorem 61 8. Topological manifolds 65 8.1. Compact manifolds 66 REFERENCES 67 Contents 1 2 ALEX GONZALEZ ELEMENTARY TOPOLOGY I 3 1. Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. What’s more, the composition of two continuous functions is also continuous. Usually, when we think of a continuous functions, the first examples that come to mind are maps f : R ! R: • the identity function, f(x) = x for all x 2 R; • a constant function f(x) = k; • polynomial functions, for instance f(x) = xn, for some n 2 N; • the exponential function g(x) = ex; • trigonometric functions, for instance h(x) = cos(x). The set of real numbers R is a natural choice of domain to begin to study more general properties of continuous functions. After all, we are familiar with many of the properties of the real line, and it is (relatively) easy to draw the graphs of functions R ! R. So, let’s review the definition of continuity for a function f : R ! R: the function f is continuous at the point a 2 R if lim f(x) = f(a) x!a (in particular we are assuming that this limit exists!). Another way of putting the above definition is the following: for each " > 0 there exists some δ > 0 such that jf(x) − f(a)j < " for all x 2 R such that jx − aj < δ: The function f is then said to be continuous on all R if it is continuous for all a 2 R. So, basically, the definition of continuity depends only on the absolute value, which is essentially a rule to measure the distance between any two numbers in R. It would seem that continuity relies on the existence of a rule to measure distances, or more precisely, a metric. 4 ALEX GONZALEZ 2. Metric spaces Let X be a set. Roughly speaking, a metric on the set X is just a rule to measure the distance between any two elements of X. Definition 2.1. A metric on the set X is a function d: X × X ! [0; 1) such that the following conditions are satisfied for all x; y; z 2 X: (M1) Positive property: d(x; y) = 0 if and only if x = y; (M2) Symmetry property: d(x; y) = d(y; x); and (M3) Triangle inequality: d(x; y) ≤ d(x; z) + d(z; y). A note of waning! The same set can be given different ways of measuring distances. Strange as it may seem, the set R2 (the plane) is one of these sets. We will see different metrics for R2 pretty soon. Example 2.2. The set X = R with d(x; y) = jx−yj, the absolute value of the difference x − y, for each x; y 2 R. Properties (M1) and (M2) are obvious, and d(x; y) = jx − yj = j(x − z) + (z − y)j ≤ jx − zj + jz − yj = d(x; z) + d(z; y): Let’s see now some examples of metrics on the set X = R2. Example 2.3. The set X = R2 with p 2 2 d (x1; x2); (y1; y2) = (x1 − y1) + (x2 − y2) 2 for each (x1; x2); (y1; y2) 2 R . Again, properties (M1) and (M2) are easy to check. Property (M3), the triangle inequality, gets its name from this example. Write x = (x1; x2); y = (y1; y2) and z = (z1; z2). Then, x d(x; y) d(x; z) y d(z; y) z Can you think of other examples of metrics on R2? If we compare the metrics in Examples 2.2 and 2.3, we can see that the metric on R2 is some sort of “extension” of the metric on R1 to a higher dimension. But we can try other ways of extending it. ELEMENTARY TOPOLOGY I 5 Example 2.4. The functions d; d0 : R2 × R2 ! [0; 1) defined by d (x1; x2); (y1; y2) = jx1 − y1j + jx2 − y2j 0 d (x1; x2); (y1; y2) = maxfjx1 − y1j; jx2 − y2jg are also metrics on R2 (the details will be checked in Examples 2.5 and 2.8 below). We have just see three different metrics on R2. But why stopping here? We can extend these metrics to Rn for all n ≥ 1. Example 2.5. Let X = Rn and let d: Rn × Rn ! [0; 1) be defined by n X d (x1; x2; : : : ; xn); (y1; y2; : : : ; yn) = jxi − yij: i=1 Then, d is a metric on Rn (sometimes known by the name of Manhattan metric). Let’s check the details: conditions (M1) and (M2) follow easily. As for the triangle inequality, we have n X d (x1; : : : ; xn);(y1; : : : ; yn) = jxi − yij = i=1 n X = j(xi − zi) + (zi − yi)j ≤ i=1 n X ≤ jxi − zij + jzi − yij = i=1 = d (x1; : : : ; xn); (z1; : : : ; zn) + d (z1; : : : ; zn); (y1; : : : ; yn) : Example 2.6. The Euclidean metric on Rn is defined by the formula v u n uX 2 d (x1; x2; : : : ; xn); (y1; y2; : : : ; yn) = t (xi − yi) ; i=1 n for each (x1; x2; : : : ; xn); (y1; y2; : : : ; yn) 2 R . As usual (M1) and (M2) are easy to check, but (M3) is not trivial at all. n Indeed, let x = (x1; x2; : : : ; xn); y = (y1; y2; : : : ; yn) and z = (z1; z2; : : : ; zn) 2 R : Let also ri = xi − zi and si = zi − yi, for i = 1; : : : ; n. We have to prove that v v v u n u n u n uX 2 uX 2 uX 2 d(x; y) = t (ri + si) ≤ t ri + t si = d(x; z) + d(z; y) (1) i=1 i=1 i=1 6 ALEX GONZALEZ Notice that both sides of the inequality are positive. Thus, by squaring the above, it is equivalent to prove that v v n n n u n u n X 2 X 2 X 2 uX 2uX 2 (ri + si) ≤ ri + si + 2t ri t si : (2) i=1 i=1 i=1 i=1 i=1 The left part of the inequality expands to n n n n X 2 X 2 X 2 X (ri + si) = ri + si + 2 risi: i=1 i=1 i=1 i=1 Replacing this in (2) and simplifying, we deduce that (1) holds if and only if n 2 n n X X 2 X 2 risi ≤ ri si : i=1 i=1 i=1 The last inequality is the Cauchy-Schwartz inequality, which we prove below as Lemma 2.7. n Lemma 2.7 (Cauchy-Schwartz inequality). For each (a1; : : : ; an); (b1; : : : ; bn) 2 R , n 2 n n X X 2 X 2 aibi ≤ ai bi : i=1 i=1 i=1 Pn Pn 2 Proof. Consider the expression i=1 j=1(aibj − ajbi) . By expanding the brackets, n n n n n n n n X X 2 X 2 X 2 X 2 X 2 X X (aibj −ajbi) = ai bj + aj bi −2 aibi ajbj : i=1 j=1 i=1 j=1 j=1 i=1 i=1 j=1 Now, collect the terms (and reindex the sums) to get n n n n n 1 X X X X X 2 (a b − a b )2 = a2 b2 − a b : 2 i j j i i i i i i=1 j=1 i=1 i=1 i=1 Since the left part of the equality is positive, this proves the statement. Example 2.8. Finally, the box metric on Rn is defined by d (x1; : : : ; xn); (y1; : : : ; yn) = maxfjxi − yij i = 1; : : : ; ng Properties (M1) and (M2) are easily seen to hold. Let’s check property (M3). Let x = n (x1; : : : ; xn); y = (y1; : : : ; yn) and z = (z1; : : : ; zn) 2 R . Then, for each i = 1; : : : ; n, jxi − yij = j(xi − zi) + (zi − yi)j ≤ jxi − zij + jzi − yij ≤ d(x; z) + d(z; y): Thus, the triangle inequality holds. As an exercise, consider the set R2 with this metric. Fix then a point a 2 R2 and draw the set 2 Da = fx 2 R d(a; x) ≤ 1g: ELEMENTARY TOPOLOGY I 7 All these examples should serve as a warning of how flexible the notion of metric is: we should not be surprised to see that statements that go against all intuition are, in fact, true. The following is a good examples of this: it is a process to define a metric on every set. This procedure is not rather descriptive, but it is nonetheless important. Example 2.9. Let X be any set.
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