top04-e09 - Prof. D. P.Patil, Department of Mathematics,...

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Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore August-December 2004 MA-231 Topology 9. Convergence — Inadquacy of sequences, Nets 1 ) —————————————————————— October 11, 2004 Eliakim Hastings Moore (1862-1932) 9.1. a). (Topology of first-countable spaces)Let X and Y be first-countable topological spaces. Then (1) A subset U X is open if and only if whenever a sequence (x n ) in X converges to a point x U , then there exists n 0 N such that x n U for all n n 0 . (2) A subset F X is closed if and only if whenever a sequence (x n ) is contained in F and x n converges to a point x X , then x F . (3) A map f : X Y is continuous if and only if whenever a sequence (x n ) converges to x in X , then the image sequence f(x n ) converges to f(x) in Y . b). Which of the above properties ain the above part a) hold for an uncountable set X with the cofinite topology ? 9.2. a). In R R , let E ={ f R R | f(x) ∈{ 0 , 1 } and f(x) = 0 only finitely often } and let g R R be the constant function 0 . Then in the product topology on R R , g E . Construct a net (f λ ) in E which converges to g . b). Let (X, d) be a mtric space and let x 0 X . Define on X \{ x 0 } by y z if and only if d(y,x 0 ) d(z,x 0 ) . With this relation X \{ x 0 } is a directed set and hence any map f : X Y into another metric space Y defines (by restricting f to X \{ x 0 } ) a net in Y . Show that this net converges to a point y 0 Y if and only if lim x x 0 f(x) = y 0 . 9.3. (Cluster points) Let X be a topological space and let x X . We say that x isa cluster point ofanet (x λ ) λ 3 in X if for each nhood U of x and for each λ 0 3 , there exists λ 3 with λ λ 0 such that x λ U . Let (x λ ) λ 3 be a net in a topological space X . a). Show that (x λ ) λ 3 has a cluster point x X if and only if (x λ ) λ 3 has a subnet which converges to x .( Proof : Suppose that x is a cluster point of the net (x λ ) . Let M : ={ (λ, U ) | λ 3, U U x with x λ U } . The set M is directed by the relation : 1 ,U 1 ) 2 ,U 2 ) if and only if λ 1 λ 2 and U 2 U 1 . The map ϕ : M 3 defined by ϕ((λ, U )) = λ is increasing and cofinal in 3 and hence defines a subnet of (x λ ) which converges to x . Conversely, suppose that ϕ : M 3 defines a subset of (x λ ) which converges to x . Then for each nhood U of x , there is some µ U M such that µ µ U implies that x ϕ(µ) U . Now, suppose that a nhood U of x and a point λ 0 3 are given. Since ϕ(M) is cofinal in 3 , there is some µ 0 M such that ϕ(µ 0 ) λ 0 . But there is also µ U M such that µ µ U implies that x ϕ(µ) U . Choose µ M such that µ µ 0 and µ µ U . Then
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.

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top04-e09 - Prof. D. P.Patil, Department of Mathematics,...

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