# Infinite sets

• Mar 11th 2011, 02:16 PM
Dreamer78692
Infinite sets
Prove that if A = R in (R; Euclidean metric) then the set A must be infinite.
Where A is a closed set.

Surely A is Infinite because it's equal to an infinite set.

How will you prove this ???(Doh)
• Mar 11th 2011, 02:19 PM
girdav
Indeed, I don't see the point. Maybe the question is : every dense subset of $\mathbb R$ must be infinite.
• Mar 11th 2011, 03:10 PM
Tinyboss
As girdav suggests...maybe the problem is asking about the closure of A in R?
• Mar 12th 2011, 11:21 PM
Dreamer78692
Sorry I made a mistake, this is the question,

Prove that if A = R in (R; Euclidean metric) then the set A' must be infinite.
Where A is a closed set.
and A' is open.

Wouldn't A, be the empty set, How can it be infinite???
• Mar 13th 2011, 04:34 AM
Plato
Quote:

Originally Posted by Dreamer78692
Sorry I made a mistake, this is the question,
Prove that if A = R in (R; Euclidean metric) then the set A' must be infinite. Where A is a closed set. and A' is open.

Even after the correction in blue above, the question still makes no sense. Are you doing a translation of the original source?

The bit in red above worries me most. If right, it seems to trivialize the question.

As for your correction, the notation $A^{\prime}$ as several different meanings in mathematics. In topology it is widely used to denote the set of all limit points of $A$, the derived set.
In some set theory textbooks the notation means the complement of the set.
I think that you must post what your text material says about $A^{\prime}$. How is it used?

Also, please review the exact wording of this problem.
• Mar 13th 2011, 04:56 AM
Dreamer78692
Also answer my question about the way the textbook defines $A^{\prime}$.