S30 - 1st December 2004 Munkres § 30 Ex. 30.3 (Morten...

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Unformatted text preview: 1st December 2004 Munkres § 30 Ex. 30.3 (Morten Poulsen). Let X be second-countable and let A be an uncountable subset of X . Suppose only countably many points of A are limit points of A and let A ⊂ A be the countable set of limit points. For each x ∈ A- A there exists a basis element U x such that x ∈ U x and U x ∩ A = { x } . Hence if a and b are distinct points of A- A then U a 6 = U b , since U a ∩ A = { a } 6 = { b } = U b ∩ A . It follows that there uncountably many basis elements, contradicting that X is second-countable. Note that it also follows that the set of points of A that are not limit points of A are countable. Ex. 30.4 (Morten Poulsen). Theorem 1. Every compact metrizable space is second-countable. Proof. Let X be a compact metrizable space, and let d be a metric on X that induces the topology on X . For each n ∈ Z + let A n be an open covering of X with 1 /n-balls. By compactness of X there exists a finite subcovering A n . Now B = S n ∈ Z + A n is countable, being a countable union of finite sets. B is a basis: Let U be an open set in X and x ∈ U . By definition of the metric topology there exists ε > 0 such that B d ( x, ε ) ⊂ U . Choose N ∈ Z + such that 2 /N < ε . Since A N covers X there exists B d ( y, 1 /N ) containing x . If z ∈ B d ( y, 1 /N ) then d ( x, z ) ≤ d ( x, y ) + d ( y, z ) ≤ 1 /N + 1 /N = 2 /N < ε, i.e. z ∈ B d ( x, ε ), hence B d ( y, 1 /N ) ⊂ B d ( x, ε ) ⊂ U . It follows that B is a basis. Ex. 30.5. Let X be a metrizable topological space. Suppose that X has a countable dense subset A . The collection { B ( a, r ) | a ∈ A, r ∈ Q + } of balls centered at points in A and with a rational radius is a countable basis for the topology: It suffices to show that for any y ∈ B ( x, ε ) there are a ∈ A and r ∈ Q + such that y ∈ B ( a, r ) ⊂ B ( x, ε ). Let r be a positive rational number such that 2 r < ε- d ( x, y ) and let a ∈ A ∩ B ( y, r ). Then y ∈ B ( a, r ), of course, and B ( a, r ) ⊂ B ( x, ε ) for if d ( a, z ) < r then d ( x, z ) ≤ d ( x, y ) + d ( y, z ) ≤ d ( x, y ) + d ( y, a ) + d ( a, z ) < d ( x, y ) + 2 r < ε ....
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This note was uploaded on 01/12/2011 for the course MATH 110 taught by Professor Brown during the Fall '08 term at Arizona Western College.

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S30 - 1st December 2004 Munkres § 30 Ex. 30.3 (Morten...

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