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Appendix a Topology of Manifolds

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Appendix A Topology of manifolds A.1 Toplogical notions A topological space is a set X together with a collection of subsets U ⊆ X called open subsets, with the following properties: • /0;X are open. • If U;U0 are open then U \U0 is open. S • For any collection Ui of open subsets, the union i Ui is open. m The collection of open subsets is called the topology of X. The space R has a standard topology given by the usual open subsets. Likewise, the open subsets of a manifold M define a topology on M. For any set X, one has the trivial topology where the only open subsets are /0and X, and the discrete topology where every subset is considered open. An open neighborhod of a point p is an open subset containing it. A topological space is called Hausdorff of any two distinct points have disjoint open neighborhoods. Any subset A ⊆ X has a subspace topology, with open sets the collection of all intersections U \ A such that U ⊆ X is open. Given a surjective map q : X ! Y, the space Y inherits a quotient topology, whose open sets are all V ⊆ Y such that the pre-image q−1(V) = fx 2 Xj q(x) 2 Vg is open. A subset A is closed if its complement XnA is open. Dual to teh staments for open sets, one has that the intersection of an arbitrary collection of closed sets is closed, and the union of two closed sets is closed. For any subset A, denote by A its closure, given as the smallest closed subset containing A. A.2 Manifolds are second countable A basis for the topology on X is a callection B = fUa g of open subsets of X such that every U is a union from sets from B. A topological space is said to be second countable if its topology has a countable basis. 1 2 A Topology of manifolds Proposition A.1. Manifolds are second countable. m Proof. The space R is second countable: A basis is given by the collection of all m rational balls, by which we mean e-balls Be (x) such that x 2 Q and e 2 Q>0. m To check it is a basis, let U ⊆ R be open, and p 2 U. Choose e 2 Q>0 such that n B2e (p) ⊆U. There exists a rational point x 2 Q with jjx− pjj < e. This then satisfies p 2 Be (x) ⊆ U. Since p was arbitrary, this proves the claim. The same reasoning m shows that for any open subset U ⊆ R , the rational e-balls that are contained in U form a basis of the topology of U. Given a manifold M, let A = f(Ua ;fa )g be a countable atlas. Then the set of −1 all fa (Be (x)), where Be (x) is a rational ball contained in fa (Ua ), is a countable basis for the topology of M. Indeed, any open subset U is a countable union over −1 all U \Ua , and each of these intersections is a countable union over all fa (Be (x)) such that Be (x) is a rational e-ball contained in U \Ua . A collection fUa g of open subsets of X is called an open covering of A ⊆ X if S A ⊆ a Ua . Consider the case A = X.A refinement of an open cover fUa g of X is an open cover fVb g of X such that each Vb is contained in some Ua . If Vb = Uf (b) for all b, then the refinement is called a subcover. A.3 Manifolds are paracompact A topological space X is called compact if every open cover of X has a finite sub- cover. A topological space is called paracompact if every open cover fUa g has a locally finite refinement fVb g: that is, every point has an open neighborhood meet- ing only finitely many Vb ’s. Proposition A.2. Manifolds are paracompact. Proof. Let fUa g be an open cover of M. The first step is to construct a sequence of open subsets W1;W2;::: such that [ Wi = M; and each Wi has compact closure with Wi ⊆ Wi+1. To this end, start with a a countable open cover O1;O2;::: of M such that each Oi has compact closure Oi. (We saw in the proof of the previous proposition how to construct such a cover.) Replacing Oi with O1 [···[ Oi we may assume O1 ⊆ O2 ⊆ ···. Define W1;W2;::: as a subsequence of O1;O2;:::, by inductively letting Wi = O j(i), with the smallest index j(i) such that W i−1 ⊆ O j(i) (the induction starts 1 with W0 := /0). 1 If M is compact, then the sequence Wi may become independent of i for large i. A.4 Partitions of unity 3 The compact subset W i+1nWi is contained in the open set Wi+2nW i−1. It is cov- ered by the open sets (Wi+2nW i−1) \ Ua , and by compactness it is already cov- ered by finitely many of those intersections. Let V (i) be this finite collection, and S (i) V = V = fVb g the open cover given as the union over all i. By construction, V = fVb g is a countable cover, which is locally finite: Given p 2 M, choose i such that p 2 Wi; by construction this Wi can only meet Vb ’s from V (k) with k ≤ i. ut Remark A.1. (Cf. Lang, page 35.) One can strengthen the result a bit, as follows: Given a cover fUa g, we can find a refinement to a cover fVb g such that each Vb is the domain of a coordinate chart (Vb ;yb ), with the following extra properties, for some 0 < r < R: (i) yb (Vb ) = BR(0), and 0 −1 (ii) M is already covered by the smaller subsets Vb = yb (Br(0)). To prove this, we change the second half of the proof as follows: For each p 2 W i+1nWi choose a coordinate chart (Vp;yp) such that yp(p) = 0, yp(Vp) = BR(0), 0 0 and Vp ⊆ (Wi+2nW i−1) \Ua . Let Vp ⊆ Vp be the pre-image of Br(0). The Vp cover (i) W i+1nWi; let V be a finite subcover and proceed as before. This remark is useful for the construction of partitions of unity. A.4 Partitions of unity We will need the following result from multivariable calculus. Lemma A.1 (Bump functions). For all 0 < r < R, there exists a function f 2 m C¥(R ), with f (x) = 0 for jjxjj ≥ R and f (x) = 1 for jjxjj ≤ r. Proof. It suffices to prove the existence of a function g 2 C¥(R) such that g(t) = 0 for t ≥ R and g(t) = 1 for t ≤ r. Indeed, given such g we can simply take f (x) = g(jjxjj). To construct g, recall that the function h(t) = 0 if t ≤ 0; h(t) = exp(−1=t) if x > 0 is smooth even at t = 0. The function h(t − r) + h(R − t) is strictly positive every- where, since for tger the first summand is positive and for t ≥ R the second sum- mand is positive. Furthermore, it agrees with h(t − r) for t ≥ R. Hence the function g 2 C¥(R) given as h(t − r) g(t) = 1 − h(t − r) + h(R −t) is 1 for t ≤ r, and 0 for t ≤ R. The support supp( f ) of a function f on M is the smallest closed subset on which teh function is non-zero. Equivalently, p 2 Mnsupp( f ) if and only if f vanishes on 4 A Topology of manifolds some open neighborhood of p. In the Lemma above, we can take f to have support 0 1 in BR(0) – simply apply the Lemma to 0 < r < R := 2 (R + r). Definition A.1. A partition of unity subordinate to an open cover fUa g of a man- ifold M is a collection of smooth functions ca , with 0 ≤ ca ≤ 1, such that supp(ca ) ⊆ Ua , and ∑ ca = 1: a Proposition A.3. For any open cover of a manifold there is a partition of unity subordinate to that cover. One can take this partition of unity to be locally finite: That is, for any p 2 M there is an open neighborhood U meeting the support of only finitely many ca ’s. Proof. Let Vb be a locally finite refinement of the cover Ua , given by coordinate 0 charts of the kind described in Remark ??, and let Vb ⊆ Vb be as decribed there. 0 Since the images of Vb ⊆ Vb are Br(0) ⊆ BR(0), we can use the Lemma above to ¥ 0 define a function fb 2 C (M) with fb (p) = 1 for p 2 Vp and supp( fb ) ⊆ Vb . Since 0 the Vb are already a cover, the sum ∑b fb is > 0) everywhere. For each index b, pick an index a such that Vb ⊆ Ua . This defines a map d : b 7! d(b) between the indexing sets. The functions ∑b2d−1(a) fb ca = : ∑g fg give the desired partition of unity..
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