Recent content by golriz

Let A be a subset of a metric space such that A ⊆ B (p, r) for some p ∈ X and r > 0. Show that diam(A) ≤ 2r. B(p,r)=(p-r,p+r) diam( B(p,r) )=sup{d(a,b)│a,b∈B(p,r) }=d(p-r,p+r)= 2r Since A ⊆ B (p, r), the diameter of A is less than the diameter of B (p, r): diam(A)≤2r Is it true and enough? I...

I use triangle inequality, so: d(x, p) + d(p, y)> d(x, y)

d(x, p) + d(p, y)> d(x, y) d(x, y)< 2r < ε ε = 2r Is it correct?

" Let X be a metric space and p be a point in X and be a positive real number. Use the definition of openness to show that the set U(subset of X) given by: U = {x∈X|d(x,p)>r} is open. " I have tried: U is open if every point of U be an interior point of U. x is an interior point of U if there...

Thank you very much!for your help

oh! at first it was typed with vague characters! but now it is in correct form

sorry! what is? " Just use that A\subseteq \overline{A}. "

Then the intersection of set of interior points of U (that equals to U, since U is open) and set of interior points of A equals to the set of interior points of the intersection of (U and A). And the set of interior points of (U and cl(A) ) equals to intersection of U and the set of interior...

But I don't know how to prove the opposite direction of this question. I have tried: using contradiction, it means the intersection of U and cl(A) is not empty.

I want to prove this theorem: " Let U be an open subset of a metric space X,and A be an arbitrary subset of X. Prove that the intersection of U and closure of A is empty if and only if the intersection of U and A be empty. " I've proved one direction of this theorem: If the intersection of U...

Is it true? " Set of interior points of the closure of A equals the set of interior points of A. "

is there any idea for continuing the above solution??

it means that the intersection of U and the closure of A is a set S which contains a point z that is in U and closure of A both. Actually I don't know how to continue the rest of the prove...

could you please help me with this question

I have proved just one direction of this question: If the intersection of U and closure of A is empty then the intersection of U and A is empty too. The closure of A is equal to the union of A and the set of all limit points(accumulation points) of A. Then we can use this definition of the...