# Infinite subsets of compact sets and limit points

Watch
Announcements

Page 1 of 1

Go to first unread

Skip to page:

*If E is an infinite subset of a compact set K, then E has a limit point in K*My proof goes like this:

Since E is an infinite set, a limit point exists. If , then so we are done. Let x be a limit point in E, such that , then . If E is a proper subset of K (otherwise , and since K is a compact set, it is closed and contains all its limit points are we are done), then .

If then since K is a compact set and is closed, is open. So x must be a interior point of , which contradicts the fact that x is a limit point of E. So and .

So is this a sound proof?

0

reply

Report

#2

(Original post by

My proof goes like this:

Since E is an infinite set, a limit point exists. If , then so we are done. Let x be a limit point in E, such that , then . If E is a proper subset of K (otherwise , and since K is a compact set, it is closed and contains all its limit points are we are done), then .

If then since K is a compact set and is closed, is open. So x must be a interior point of , which contradicts the fact that x is a limit point of E. So and .

So is this a sound proof?

**0x2a**)*If E is an infinite subset of a compact set K, then E has a limit point in K*My proof goes like this:

Since E is an infinite set, a limit point exists. If , then so we are done. Let x be a limit point in E, such that , then . If E is a proper subset of K (otherwise , and since K is a compact set, it is closed and contains all its limit points are we are done), then .

If then since K is a compact set and is closed, is open. So x must be a interior point of , which contradicts the fact that x is a limit point of E. So and .

So is this a sound proof?

"Since E is an infinite set, a limit point exists." Why?

"K is compact implies K is closed." This is false: consider the cofinite topology on Z. All subsets of Z are compact, but only the finite ones (and Z) are closed. (It is true if K is also Hausdorff.)

0

reply

(Original post by

As I read this, I thought of the following things:

"Since E is an infinite set, a limit point exists." Why?

"K is compact implies K is closed." This is false: consider the cofinite topology on Z. All subsets of Z are compact, but only the finite ones (and Z) are closed. (It is true if K is also Hausdorff.)

**Smaug123**)As I read this, I thought of the following things:

"Since E is an infinite set, a limit point exists." Why?

"K is compact implies K is closed." This is false: consider the cofinite topology on Z. All subsets of Z are compact, but only the finite ones (and Z) are closed. (It is true if K is also Hausdorff.)

0

reply

Report

#4

(Original post by

Oh I forgot to mention that where is a metric space. Does that make any difference? This is from an analysis book and not a topology one

**0x2a**)Oh I forgot to mention that where is a metric space. Does that make any difference? This is from an analysis book and not a topology one

0

reply

(Original post by

That justifies "K is closed". I'm still not completely sure about "a limit point exists" - do you have a justification for that?

**Smaug123**)That justifies "K is closed". I'm still not completely sure about "a limit point exists" - do you have a justification for that?

Oh and if E has a limit point , then is a limit point of K, and since K is closed, K must contain all its limit points, so . Would this be a valid proof once I validate that a limit point of E exists at all?

0

reply

Report

#6

(Original post by

Well I'm stuck but if E has not limit points, then the set of limit points of E is the empty set, and so E is closed. Since E is a subset of a compact set and is closed, E is also a compact set.

Oh and if E has a limit point , then is a limit point of K, and since K is closed, K must contain all its limit points, so . Would this be a valid proof once I validate that a limit point of E exists at all?

**0x2a**)Well I'm stuck but if E has not limit points, then the set of limit points of E is the empty set, and so E is closed. Since E is a subset of a compact set and is closed, E is also a compact set.

Oh and if E has a limit point , then is a limit point of K, and since K is closed, K must contain all its limit points, so . Would this be a valid proof once I validate that a limit point of E exists at all?

0

reply

(Original post by

Actually, a limit point of any nonempty set exists, doesn't it? Take the sequence (x, x, …); this has limit x. That makes the original theorem completely trivial. I've presumably badly misunderstood something.

**Smaug123**)Actually, a limit point of any nonempty set exists, doesn't it? Take the sequence (x, x, …); this has limit x. That makes the original theorem completely trivial. I've presumably badly misunderstood something.

0

reply

X

Page 1 of 1

Go to first unread

Skip to page:

### Quick Reply

Back

to top

to top