Recent content by sazanda

totally bounded how Can I show that the closure of a totally bounded set is (totally) bounded? solution Tried: Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S let x be a limit points of S that is in S closure...

Homework Statement Let K be a subset of R. Prove that if every sequence in K has an accumulation point, then K must be compact. Homework Equations I tried to proof it below. Am I on the right track? The Attempt at a Solution My intuition; Let x_n be sequence in K whose...

A1={[0,1/3],[2/3,1]} A2={[0,1/9],[2/9,3/9],[6/9,7/9],[8/9,9/9]} : : : intersection of all Ai is the Cantor set. This is definition that Prof. defined.

Let me clarify myself. let X be a collection of disjoint closed sets. Define X := { {x} such that x in ℝ } {x}_1 is the one of the disjoint closed set. {x}_2 is another disjoint closed set. and so fourth {x}_i is the another disjoint closed set Since ℝ is uncountable X must be...

Homework Statement Determine whether the following statements are true or false a) Every pairwise disjoint family of open subsets of ℝ is countable. b) Every pairwise disjoint family of closed subsets of ℝ is countable. Homework Equations part (a) is true. we can find 1-1...

a cantor set

I need some kind of initiation. Prof. had just defined the Cantor set and assigned this problem. I do not have more info about this. I looked the internet they are little bit complicated.

Homework Statement Let C be a Cantor set and let x in C be given prove that a) Every neighborhood of x contains points in C, different from x. b) Every neighborhood of x contains points not in C Homework Equations How can I start to prove? The Attempt...

We didn't cover totally boundedness. I think we should use definition of precompactess.

Definition: A set S is precompact if every ε>0 then S can be covered by finitely many discs of radius ε .

Homework Statement Let S be a subset of C. Prove that S is precompact if and only if S(closure) is compact. Homework Equations I have already showed if S(closure) compact, then S is precompact how can I show if S is precompact, then S(closure) is compact? The Attempt at a Solution