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Some Results for Locally Compact Hausdorff Spaces

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Some Results for Locally Compact Hausdorff Spaces Some results for locally compact Hausdorff spaces Shiu-Tang Li Finished: April 21, 2013 Last updated: November 2, 2013 The following materials are all taken from [Rudin]. Other references: [Munkres]; [Folland] sec 4.5; [Richard Bass] sec 20.9 1 General Stuff Theorem 1.1. Suppose U is open in a locally compact Hausdorff space X, K ⊂ U, and K is compact. Then there is an open set V with compact closure s.t. K ⊂ V ⊂ V ⊂ U. [Rudin p.37] Theorem 1.2. (Urysohn's lemma) Suppose X is a locally compact Hausdorff space, V is open in X, K ⊂ V , and K is compact. Then there exists an f 2 Cc(X) s.t. χK ≤ f ≤ χV . [Rudin p.39] Remark 1.3. See also [Munkres] for another version of Urysohn's lemma, which assumes that X is normal. The arguments are also slightly different. Theorem 1.4. (Partition of unity) Suppose V1; ··· ;Vn are open subsets of a locally compact Hausdorff space X, K is compact, and K ⊂ V1 [···[ Vn. Then there exists 0 ≤ hi < χVi , 1 ≤ i ≤ n, so that h1(x) + ··· + hn(x) = 1 for all x 2 K. [Rudin p.40] Remarks 1.5. (1) Must take a look at the original proof to see if hi is continuous and compact supported. See also [245B, Notes 12: Continuous functions on locally compact Hausdorff spaces] of Terence Tao's website. (2) Theorem 1. ) Theorem 2. ) Theorem 4. (3) See [Munkres] page 225. for 1 another version of partition of unity (on normal spaces). Remarks 1.6. I skip the Riesz representation theorem. Theorem 1.7. (Lusin's theorem) Suppose X is a locally compact Hausdorff space, µ is a Radon measure on X, and µ is complete. If f is a complex measurable function on X, and there exists A s.t. µ(A) < 1, f(x) = 0 for x2 = A, then, given > 0, there exists some g 2 Cc(X) so that µ(g 6= f) < , and we may arrange it so that supx2X jg(x)j ≤ supx2X jf(x)j. [Rudin p.55] Remark 1.8. Wikipedia says Lusin's theorem holds for second countable topological space. (?) Theorem 1.9. Suppose X is a locally compact Hausdorff space, µ is a Radon measure on X, and µ is complete. For 1 ≤ p < 1, Cc(X) is dense in Lp(µ). [Rudin p.69] Definition 1.10. Let X be a locally compact Hausdorff space. A complex function f is said to vanish at infinity if for every > 0, there exists a compact set K ⊂ X s.t. jf(x)j < for all x 2 Kc. The class of all continu- ous function f on X which vanish at infinity is called C0(X). [Rudin p.70] Theorem 1.11. Suppose X is a locally compact Hausdorff space. C0(X) is the completion of Cc(X) w.r.t. supremum norm. [Rudin p.70] 2 One-point compactification Theorem 2.1. Let (X; T ) be a Hausdorff space. Let 1 be a point not in X, and let X∗ := X [ f1g. Define T ∗ := T [ T 0, where T 0 := fU ⊂ X∗ : X∗ n U compact in (X; T )g . We claim that T ∗ is a topology for X∗. Proof. (1) ? 2 T ⊂ T ∗. Since X∗ n X∗ = ?, which is compact in (X; T ), X∗ 2 T 0 ⊂ T ∗. 0 (2) Let fUαgα2I ⊂ T . We may write each Uα = (X n Kα) [ f1g, S S Kα is compact in (X; T ). We have α2I Uα = α2I (X n Kα) [ f1g = T 0 (X n α2I Kα) [ f1g, which is still in T (The intersection of compacts is a 0 compact set since X is Hausdorff). For A1 2 T , A2 = (X n K) [ f1g 2 T , 2 0 ∗ A1 [ A2 = X n ((X n A1) \ K) 2 T ⊂ T , where (X n A1) \ K is compact again because X is Hausdorff. 0 (3) Let U1; ··· ;Un 2 T , and we write each Uj = (X n Kj) [ f1g, Kj Tn Tn is compact in (X; T ). Therefore, j=1 Uj = j=1(X n Kj) [ f1g = (X n Sn 0 ∗ 0 j=1 Kj) [ f1g 2 T ⊂ T . For A1 2 T , A2 = (X n K) [ f1g 2 T , ∗ A1 \ A2 = (X n K) \ A1 2 T ⊂ T . When (X; T ) is a locally compact Hausdorff space, we have the following result: Theorem 2.2. Let (X; T ) be a locally compact Hausdorff space. Then the one-point compactification (X∗;T ∗) as shown in Theorem 2.1 is a com- pact Hausdorff space. [Richard Bass, real analysis, Sec 20.9] Remark 2.3. (X; T ) coincides with the subspace topology induced from (X∗;T ∗). Theorem 2.4. Let (X; T ) be a locally compact Hausdorff space. Then every function f : E ! R belongs to C0(X) if and only if the extension f~ : X∗ ! R, f~(1) = 0, f~(x) = f(x) for x 2 X is a continuous function on (X∗;T ∗). ~−1 Proof. Let f 2 C0(X). For any V open in R, if 0 2= V , then f (V ) = f −1(V ) 2 T . If 0 2 V , then f~−1(V ) n f1g = f −1(V ) ⊃ X n K for some com- pact set K in (X; T ). It follows that X n f −1(V ) is a compact set in (X; T ) (since it is a closed subset of K). As a result, f~−1(V ) = f1g [ f −1(V ) = f1g [ (X n (X n f −1(V ))) 2 T 0 ⊂ T ∗, which proves f~ 2 C(X∗). Conversely, let f~ 2 C(X∗). For any V open in R, if 0 2= V , then f −1(V ) = f~−1(V ) 2 T . If 0 2 V , then f~−1(V ) = (X n K) [ f1g, K compact in (X; T ), and thus f −1(V ) = X n K 2 T . This proves f is continuous on (X; T ), and actually, that f −1(V ) = X n K for 0 2 V is exactly equivalent to the fact that f vanishes at infinity. Theorem 2.5. Let (X; T ) be a locally compact Hausdorff second count- able space (LCCB). Then the one-point compactification (X∗;T ∗) as shown in Theorem 2.1 is second countable. Proof. We first notice that the collection C of all open sets with compact closure form a basis for T . To see C is a basis, first, from the definition of locally compact space we know that every x 2 X is covered by some open set 3 in C . Second, for C1;C2 2 C , and x 2 C1 \C2, there exists some open neigh- borhood of U of x so that U ⊂ C1 [ C2, and we have U ⊂ C1 [ C2 = C1 [ C2. This implies U is compact, and U 2 C . To see C generates T , for any open set U that contains x, there exists some V with compact closure s.t. x 2 V ⊂ V ⊂ U (Theorem 1.1). Since (X; T ) is second countable, every basis has a countable subfamily that is still a basis for T . Therefore, we may apply this result to C above to get a countable basis B := fB1;B2; · · · g, where each Bj 2 B has compact closure. ∗ 0 0 SN Define B := B [ B , where B := f(X n j=1 Bnj ) [ f1g : fBnj gj ∗ ∗ is a subsequence of fBngn;N 2 Ng. To see B is a basis in X , first we note that for every x 2 X∗ is covered by some B 2 B∗. Second, for 0 0 B1;B2 2 B (resp: B ); x 2 B1 \ B2, there exists some B3 2 B (resp: B ) 0 so that x 2 B3 ⊂ B1 \ B2. For B1 2 B, B2 2 B , B1 \ B2 is an open set in X, so there exists some B3 2 B so that x 2 B3 ⊂ B1 \ B2 (Since B is a basis for (X; T )). To see B∗ generates T ∗, the only nontrivial fact to prove is that for any open set U in T ∗ of form (X n K) [ f1g, K compact in (X; T ), x = f1g 2 U, there exists some B 2 B0 so that x 2 B ⊂ U. This is due to the fact that since K is compact in (X; T ), we may find some fBm1 ; ··· ;Bmk g ⊂ B that covers K, and it follows that Sk x 2 (X n j=1 Bmj ) [ f1g ⊂ U. Therefore, B∗ is a countable basis for (X∗;T ∗), and the proof is complete. Corollary 2.6. Let (X; T ) be a locally compact Hausdorff second count- able space (LCCB). Then X is metrizable. Proof. By the previous theorems, the one-point compactification (X∗;T ∗) is second countable and compact Hausdorff. Since for compact Hausdorff spaces, second countability is equivalent to metrizability, (X∗;T ∗) is metriz- able. Therefore, as a subspace of a metric space, (X; T ) is also a metrizable topological space. 4.
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