Self explanatory really. Couldn't find an answer. I know that there can be sets with a highest and lowest element that are dense in themselves for example
$\mathbb{Q}\cap[0,1]$
but I'm not sure about sets with just a single element.
Asked
Active
Viewed 219 times
4
-
2Does every nonempty open set intersect ${0}$? – pancini Oct 11 '19 at 04:43
-
As the previous commenter noted, it’s all tied to the definition. Since it’s a single set, even if you put the trivial topology on it, you still have your element in an open set, moreover, every open set that is non empty intersects that set, i. e. Itself. – Elen Khachatryan Oct 11 '19 at 04:56
-
@ElenKhachatryan i'm a little confused – John Doe Oct 11 '19 at 05:37
-
@ElenKhachatryan the trivial topology is the only topology on a singleton. – Vsotvep Oct 11 '19 at 06:00
-
Are we talking about topological density or order density? – Cheerful Parsnip Oct 11 '19 at 06:12
-
I think the definition of dense using isolated points is incorrect exactly for singleton sets. – Ingix Oct 11 '19 at 07:10
2 Answers
4
A subset $X\subset A$ of a topological space $A$ is dense in itself if $X$ contains no isolated points.
An isolated point is any point $x\in X$ such that $\{x\}$ is open in $X$ (with the subspace topology).
So following this definition, if $X$ is a singleton, then $X$ is not dense in itself, since $X$ is open in $X$.
Vsotvep
- 6,123
2
Thanks to Vsotvep to pointing out my mistake. I'll let this (modified) answer stand because maybe sombody else does the same mistake as I with an unknown definition that reads dangerously close to a more well known one.
If $X$ is a topological space and A subset of X, then Wikipedia gives a dense set definition of A is dense in X:
Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A (i.e., A has non-empty intersection with every non-empty open subset of X).
Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty.
So looking at both equivalent definitions it should be clear that if $A=X=\{0\}$, then A is dense in X.
But Wikipedia also gives dense in itself definition for some subset $A$ of a given topological space $X$:
In mathematics, a subset A of a topological space is said to be dense-in-itself if A contains no isolated points.
Using that definition, which is the one appropriate one for this question, as Vsotvep showed in their answer, $A$ is not dense in itself.
Ingix
- 14,494
-
This was my original answer, until I discovered that dense-in-itself is something different (and less trivial): https://en.wikipedia.org/wiki/Dense-in-itself – Vsotvep Oct 11 '19 at 07:17
-