exer2Math420 - Show that if X is an infinite set, it is...

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Math 420, Point Set Topology Suggested exercises 2 Problem 1. Let ¯ ρ be the uniform metric on R ω . Given x = { x n } ∈ R ω and ε (0 , 1), let U ( x,ε ) = ( x 1 + ε,x 1 - ε ) × ··· × ( x n + ε,x n - ε ) × ... a. Show that U ( x,ε ) is not equal to the ε -ball B ¯ ρ ( x,ε ). b. Show that U ( x,ε ) is not open in the uniform topology. c. Show that B ¯ ρ ( x,ε ) = S δ<ε U ( x,δ ). Problem 2. Consider the box, product and uniform topology on R ω . In which topologies do the following sequences converge? x 1 = (1 , 1 , 0 , 0 ,... ) x 2 = ( 1 2 , 1 2 , 0 , 0 ,... ) x 3 = ( 1 3 , 1 3 , 0 , 0 ,... ) ... y 1 = (1 , 1 , 1 , 1 ,... ) y 2 = (0 , 2 , 2 , 2 ,... ) y 3 = (0 , 0 , 3 , 3 ,... ) ... Problem 3. Show that the countable product of metrizable spaces is metriz- able. Problem 4. Show that if Q X α is connected and nonempty, then each X α is connected. Problem 5.
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Unformatted text preview: Show that if X is an infinite set, it is connected in the topology J = { A : X \ A is finite or all of X } . Problem 6. Is the space R l connected? Problem 7. Is the product of path connected spaces is path connected? Problem 8. If A is a connected subset of X , is the interior of A is connected? Does the converse hold? Problem 9. Show that if U is open path connected subset of R 2 then U is path connected. Hint: Show that given X ∈ U , the set of points x ∈ U that can be joined to x by a path in U is both open and closed. 1...
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This note was uploaded on 11/14/2011 for the course MATH 367 taught by Professor Sdd during the Spring '11 term at Middle East Technical University.

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