Closed, Bounded but not Compact.

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November 6th 2010, 03:58 PM
matt.qmar

Closed, Bounded but not Compact.

The set of rationals Q forms a metric spce by $d(p,q) = | p - q |$

Then a subset E of Q is defined by $E = \{ p \in Q : 2 < p^2 < 3 \}$
So I am trying to show that E is closed and bounded, but not compact.

To me, it is clear than E is bounded (by 2 and 3?!).

I am having trouble showing E is closed. I believe 2 and 3 are limit points of E but E does not contain them, so E is not closed? also, E does not contain e = 2.718281828.... but that is also a limit point of E, is it not? Maybe I am confused with my definition of limit point.... are any of these (2, 3, e) actually limit points of E?

I think I can show E is not compact, since the open cover of E,
$\bigcup\ (2 + \frac{1}{n}, 3 - \frac{1}{n})$ for n = 2, 3, 4, ....
has no finite subcover?

I am worried I am perhaps misunderstanding some definitions... Any help or direction would be greatly appriciated.

Thank you in advanced for you time!
November 6th 2010, 04:27 PM
Plato

Do you understand that $E = \left( {\sqrt 2 ,\sqrt 3 } \right) \cap \mathbb{Q}~?$

Consider this cover $O_n = \left( {\sqrt 2 + \frac{{\sqrt 3 - \sqrt 2 }}{{3n}},\sqrt 3 - \frac{{\sqrt 3 - \sqrt 2 }}{{3n}}} \right)$
November 6th 2010, 04:35 PM
matt.qmar

Wow, thank you for pointing that out. I somehow goofed on what the set E was!

However, even in this new light, E does not appear to be closed... $\sqrt 2$ is not in E, but it is a limit point of E?

Thanks!
November 6th 2010, 05:09 PM
Plato

You are still missing the point $\sqrt2\notin \mathbb{Q}$ .
Does the set contain all of its limit points in $\mathbb{Q}~?$
November 6th 2010, 05:44 PM
Drexel28

Quote:

Originally Posted by matt.qmar

The set of rationals Q forms a metric spce by $d(p,q) = | p - q |$

Then a subset E of Q is defined by $E = \{ p \in Q : 2 < p^2 < 3 \}$
So I am trying to show that E is closed and bounded, but not compact.

To me, it is clear than E is bounded (by 2 and 3?!).

I am having trouble showing E is closed. I believe 2 and 3 are limit points of E but E does not contain them, so E is not closed? also, E does not contain e = 2.718281828.... but that is also a limit point of E, is it not? Maybe I am confused with my definition of limit point.... are any of these (2, 3, e) actually limit points of E?

I think I can show E is not compact, since the open cover of E,
$\bigcup\ (2 + \frac{1}{n}, 3 - \frac{1}{n})$ for n = 2, 3, 4, ....
has no finite subcover?

I am worried I am perhaps misunderstanding some definitions... Any help or direction would be greatly appriciated.

Thank you in advanced for you time!

Another simplistic theorem tells you that if $X,Y,Z$ are metric spaces then $Z$ is a compact subspaces of $Y$ if and only if $Z$ is a compact subspaces of $Y$ . Note though that with this in hand $E$ will be a compact subspaces of $\mathbb{Q}$ if and only if it's a compact subspace of $\mathbb{R}$ . But, considering that it isn't closed ( $\sqrt{2}$ is a limit point not in the set) it follows that $E$ is not compact.