- #1

- 294

- 1

## Main Question or Discussion Point

Are there any down to earth examples besides the empty set?

Edit: No discrete metric shenanigans either.

Edit: No discrete metric shenanigans either.

Last edited:

- Thread starter Poopsilon
- Start date

- #1

- 294

- 1

Are there any down to earth examples besides the empty set?

Edit: No discrete metric shenanigans either.

Edit: No discrete metric shenanigans either.

Last edited:

- #2

- 360

- 0

Actually, I think any disk in [itex] \mathbb{Q}_p [/itex] would work.

I'm not sure if you'd count that as "down-to-earth."

- #3

- 382

- 4

- #4

- 294

- 1

I'm confused, so there are such open compact sets, but only in non-connected metric spaces?

- #5

- 360

- 0

I think that's correct. Suppose we have a nonempty proper subset S of a metric space X with S open and compact. As deluks said, then S is closed, hence clopen. Then S and [itex] X \setminus S [/itex] are both open, disjoint, and nonempty and [itex] X = S \cup (X \setminus S)[/itex]. Therefore X is disconnected.I'm confused, so there are such open compact sets, but only in non-connected metric spaces?

Incidentally, the p-adic numbers are totally disconnected.

- #6

- 382

- 4

- #7

disregardthat

Science Advisor

- 1,854

- 33

- #8

HallsofIvy

Science Advisor

Homework Helper

- 41,808

- 933

- #9

- 22,097

- 3,279

Only in Hausdorff spaces. There are non-Hausdorff spaces with compact sets which are not closed.A compact subset of a metric space is closed.

For example, take X an infinite set and equip it with the finite complement topology:

[tex]\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is finite}\}[/tex]

then all subsets of X are compact, and so are the open subsets. But of course, not all subsets are closed.

This is an example of a noetherian space (= a space in which all open subsets are compact). They arise naturally in algebraic geometry.

EDIT: I'm sorry, I saw now that they were asking for a metric space. But I'll leave my answer here because it might be useful to some.

- #10

Bacle2

Science Advisor

- 1,089

- 10

"A compact subset of a metric space is closed"

Just a quick comment, to refresh my metric topology: an argument for why compact subsets X of metric spaces M are closed: as metric subspaces, compact metric spaces are complete. Now, use closed set version that a subset is closed iff (def.) it contains all its limit points. Now, assume there is a limit point m of X that is not in X. Then you can construct a Cauchy sequence in X that would converge to m, e.g., by taking balls B(m, 1/2

- #11

- 22,097

- 3,279

Indeed. Do notice that you have used here that limits of sequences are unique!! The argument does not hold for pseudo-metric spaces.

"A compact subset of a metric space is closed"

Just a quick comment, to refresh my metric topology: an argument for why compact subsets X of metric spaces M are closed: as metric subspaces, compact metric spaces are complete. Now, use closed set version that a subset is closed iff (def.) it contains all its limit points. Now, assume there is a limit point m of X that is not in X. Then you can construct a Cauchy sequence in X that would converge to m, e.g., by taking balls B(m, 1/2^{n}), but then X cannot be complete, since m is not in X.

- Last Post

- Replies
- 10

- Views
- 6K

- Last Post

- Replies
- 3

- Views
- 3K

- Last Post

- Replies
- 3

- Views
- 2K

- Replies
- 2

- Views
- 3K

- Replies
- 12

- Views
- 9K

- Replies
- 1

- Views
- 1K

- Replies
- 12

- Views
- 5K

- Last Post

- Replies
- 1

- Views
- 5K

- Last Post

- Replies
- 2

- Views
- 1K

- Last Post

- Replies
- 4

- Views
- 2K