Homework Statement
Show that
If S is totally bounded in ℂ, then the S closure is also totally bounded in ℂ.
Homework Equations
The Attempt at a Solution
Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S...
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...
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...
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...
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