What is a countable dense subset?

What is a countable dense subset?

Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.

Is every dense subset of the real line is countable?

A dense set of the reals is always infinite. It need not be countable: For example, the reals are dense in themselves. And a dense set needs not be infinite either. For example, {1} is a finite set that is dense in itself.

Are continuous functions dense?

So continuous functions are dense in the step functions, and hence, Lp. 2n+1 , 1 2n ]. f ∈ L∞, but ||f −s||∞ ≥ 1/2 for any step function s. functions are continuous.

Are dense sets countable?

The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself.

How do you prove a countable dense subset?

Proof. If (Un)n∈N is a countable basis of open sets of a metrizable space E, and if un is an element of Un, then the set of the un forms a dense subset of E. Conversely, let D be a countable dense subset of a metric space E.

What is dense set in real analysis?

A subset S ⊂ X S \subset X S⊂X is called dense in X if any real number can be arbitrarily well-approximated by elements of S. For example, the rational numbers Q are dense in R, since every real number has rational numbers that are arbitrarily close to it.

How do you prove dense subsets?

Let be a metric space. A set Y ⊆ X is called dense in if for every x ∈ X and every , there exists y ∈ Y such that . d ( x , y ) < ε . In other words, a set Y ⊆ X is dense in if any point in has points in arbitrarily close.

Are continuous functions dense in l2?

X→C X → ℂ with compact support. Theroem – For every 1≤p<∞ 1 ≤ p < ∞ , Cc(X) ⁢ is dense in Lp(X) ⁢ (http://planetmath.org/LpSpace)….compactly supported continuous functions are dense in Lp.

Are smooth functions dense in LP?

Yes. In fact, by the Stone-Weierstrass theorem and the existence of smooth bump functions, smooth functions with compact support are uniformly dense in the space of continuous functions with compact support.

What are the dense subsets of a discrete metric space?

In a discrete space, the singleton set {x} is open. The only way this set can have non-empty intersection with D is if we have x∈D. But this means that the only dense subspace of a discrete space X is X itself. Hence, the only way to have a countable dense subset of a discrete space is if the space itself is countable.

What is dense set in mathematics?

Definition 2.1. A set Y ⊆ X is called dense in if for every x ∈ X and every , there exists y ∈ Y such that . d ( x , y ) < ε . ? In other words, a set Y ⊆ X is dense in if any point in has points in arbitrarily close.

What is everywhere dense set?

A subset A of a topological space X is dense for which the closure is the entire space X (some authors use the terminology everywhere dense). A common alternative definition is: a set A which intersects every nonempty open subset of X.