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The Topology of Subsets of Rn

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The Topology of Subsets of Rn APPENDIXA The Topology of Subsets of Rn In this appendix, we briefly review some notions from topology that are used throughout the book. The exposition is intended as a quick review for readers with some previous exposure to these topics. 1. Open and Closed Sets and Limit Points The natural distance function on Rn is defined such that for all a, b ∈ Rn, dist(a, b)=|a − b| = a − b, a − b. Its most important property is the triangle inequality: Proposition A.1 (TheTriangleInequality). For all a, b, c ∈ Rn, dist(a, c) ≤ dist(a, b)+dist(b, c). Proof. The Schwarz inequality (Lemma 1.12) says that |v, w| ≤ |v||w| for all v, w ∈ Rn.Thus, |v + w|2 = |v|2 +2v, w + |w|2 ≤|v|2 +2|v|·|w| + |w|2 =(|v| + |w|)2. © Springer International Publishing Switzerland 2016 345 K. Tapp, Differential Geometry of Curves and Surfaces, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-39799-3 346 A. THE TOPOLOGY OF SUBSETS OF Rn So |v + w|≤|v| + |w|. Applying this inequality to the vectors pictured in Fig. A.1 proves the triangle inequality. Figure A.1. Proof of the triangle inequality Topology begins with precise language for discussing whether a subset of Euclidean space contains its boundary points. First, for p ∈ Rn and r>0, we denote the ball about p of radius r by B(p,r)={q ∈ Rn | dist(p, q) <r}. In other words, B(p,r) contains all points closer than a distance r from p. Definition A.2. Apointp ∈ Rn is called a boundary point of a subset S ⊂ Rn if for all r>0,theballB(p,r) contains at least one point in S and at least one point not in S. The collection of all boundary points of S is called the boundary of S. Sometimes, but not always, boundary points of S are contained in S. For example, consider the “open upper half-plane” H = {(x, y) ∈ R2 | y>0}, and the “closed upper half-plane” H = {(x, y) ∈ R2 | y ≥ 0}. The x-axis, {(x, 0) | x ∈ R2}, is the boundary of H and also of H.SoH contains none of its boundary points, while H contains all of its boundary points. This distinction is so central that we introduce vocabulary for it: Definition A.3. Let S ⊂ Rn be a subset. (1) S is called open if it contains none of its boundary points. (2) S is called closed if it contains all of its boundary points. In the previous example, H is open, while H is closed. If part of the x-axisisadjoinedtoH (say the positive part), the result is neither closed nor open, since it contains some of its boundary points but not all of them. AsetS ⊂ Rn and its complement Sc = {p ∈ Rn | p ∈/ S} clearly have the same boundary. If S contains none of these common boundary points, 1. OPEN AND CLOSED SETS AND LIMIT POINTS 347 then Sc must contain all of them, and vice versa. So we learn a relationship between a subset and its complement: Lemma A.4. AsetS ⊂ Rn is closed if and only if its complement, Sc,isopen. The following provides a useful alternative definition of “open”: Lemma A.5. AsetS ⊂ Rn is open if and only if for all p ∈ S,thereexistsr>0 such that B(p,r) ⊂ S. Proof. If S is not open, then it contains at least one of its boundary points, and no ball about such a boundary point is contained in S.Con- versely, suppose that there is a point p ∈ S such that no ball about p is contained in S.Thenp is a boundary point of S,soS is not open. The proposition says that if you live in an open set, then so do all of your sufficiently close neighbors. How close is sufficient depends on how close you live from the boundary. For example, the set S =(0, ∞) ⊂ R is open because for every x ∈ S, the ball B(x, x/2) = (x/2, 3x/2) lies inside of S;seeFig.A.2.Whenx is close to 0, the radius of this ball is small. Similarly, for every p ∈ Rn and r>0, the ball B = B(p,r)isitselfopen, r−dist(p,q) because about every q ∈ B, the ball of radius 2 lies in B by the triangle inequality; see Fig. A.3. Figure A.2. The set (0, ∞) ⊂ R is open because it contains a ball about each of its points Figure A.3. The set B(p,r) ⊂ Rn is open because it con- tains a ball about each of its points 348 A. THE TOPOLOGY OF SUBSETS OF Rn Lemma A.6. The union of a collection of open sets is open. The intersection of a finite collection of open sets is open. The intersection of a collection of closed sets is closed. The union of a finite collection of closed sets is closed. The collection of all open subsets of Rn is called the topology of Rn.It is surprising how many important concepts are topological, that is, definable purely in terms of the topology of Rn. For example, the notion of whether a subset is closed is topological. The distance between points of Rn is not topo- logical. The notion of convergence is topological by the second definition below, although it may not initially seem so at first: Definition A.7. n An infinite sequence {p1, p2,...} of points of R is said to converge to p ∈ Rn if either of the following equivalent conditions holds: (1) limn→∞ dist(p, pn)=0. (2) For every open set, U, containing p, there exists an integer N such that pn ∈ U for all n>N (Fig. A.4). Figure A.4. A convergent sequence is eventually inside of every open set containing its limit Definition A.8. Apointp ∈ Rn is called a limit point of a subset S ⊂ Rn if there exists an infinite sequence of points of S that converges to p. Every point p ∈ S is a limit point of S, as evidenced by the redundant infinite sequence {p, p, p,...}. Every point of the boundary of S is a limit point of S as well. In fact, the collection of limit points of S equals the union of S and the boundary of S. Therefore, asetS ⊂ Rn is closed if and only if it contains all of its limit points, since this is the same as requiring it to contain all of its boundary points. It is possible to show that a sequence converges without knowing its limit, just by showing that the terms get closer and closer to each other: 1. OPEN AND CLOSED SETS AND LIMIT POINTS 349 Definition A.9. n An infinite sequence of points {p1, p2,...} in R is called a Cauchy sequence if for every >0, there exists an integer N such that dist(pi, pj ) <for all i, j > N. It is straightforward to prove that every convergent sequence is Cauchy. A fundamental property of Euclidean space is the converse: Proposition A.10. Every Cauchy sequence in Rn converges to some point of Rn. We end this section with an important relative notion of open and closed: Definition A.11. Let V ⊂ S ⊂ Rn be subsets. (1) V is called open in S ifthereexistsanopensubsetofRn whose intersection with S equals V. (2) V is called closed in S if there exists a closed subset of Rn whose intersection with S equals V. For example, setting S2 = {(x, y, z) ∈ R3 | x2 + y2 + z2 =1}, notice that the hemisphere {(x, y, z) ∈ S2 | z>0} is open in S2, because it is the intersection with S2 of the following open set: {(x, y, z) ∈ R3 | z>0}. It is straightforward to show that V is open in S if and only if {p ∈ S | p ∈/ V } is closed in S. The following is a useful equivalent characterization of “open in” and “closed in”: Proposition A.12. Let V ⊂ S ⊂ Rn be subsets. (1) V is open in S if and only if for all p ∈ V, there exists r>0 such that {q ∈ S | dist(p, q) <r}⊂V. (2) V is closed in S if and only if every p ∈ S that is a limit point of V is contained in V. Part(1)saysthatifyouliveinasetthat’sopeninS,thensodoallof your sufficiently close neighbors in S. Part (2) says that if you live in a set that’s closed in S, then you contain all of your limit points that belong to S. For example, the interval [0, 1) is neither open nor closed in R, but is open in [0, 2] and is closed in (−1, 1). Let p ∈ S ⊂ Rn.Aneighborhood of p in S means a subset of S that is open in S and contains p. For example, (1 − , 1+) is a neighborhood of 1in(0, 2) for every ∈ (0, 1]. Also, [0,) is a neighborhood of 0 in [0, 1] for every ∈ (0, 1]. 350 A. THE TOPOLOGY OF SUBSETS OF Rn The collection of all subsets of S that are open in S is called the topology of S. It has many natural properties. For example, the relative version of Lemma A.6 is true: the union of a collection of subsets of S that are open in S is itself open in S, and similarly for the other statements. In the remainder of this appendix, pay attention to which properties of a set S are topological, that is, definable in terms of only the topology of S.
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