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

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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). (T o p o l o g y o f f i r s t - c o u n t a b l e s p a c e s ) 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. ( C l u s t e r p o i n t s ) Let X be a topological space and let x X . We say that x is a c l u s t e r point of a net (x λ ) λ in X if for each nhood U of x and for each λ 0 , there exists λ with λ λ 0 such that x λ U . Let (x λ ) λ be a net in a topological space X . a). Show that (x λ ) λ has a cluster point x X if and only if (x λ ) λ 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 , 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 defined by ϕ((λ, U)) = λ is increasing and cofinal in and hence defines a subnet of (x λ ) which converges to x . Conversely, suppose that ϕ : M 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 are given. Since ϕ(M) is cofinal in , 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 ϕ(µ ) = λ λ 0 , since ϕ(µ ) ϕ(µ 0 ) , and x λ = x ϕ(µ ) U , since µ µ U . Therefore for any nhood U of x and any λ 0 , there is some λ λ 0 with x λ U and hence it follows that x is a cluster point of the net (x λ ) .

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