Denseness: Difference between revisions
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==Formal definition== | ==Formal definition== | ||
Let ''X'' be a [[topological space]]. A subset <math>\scriptstyle A \subset X</math> is said to be '''dense''' in ''X'', or to be a dense set in ''X'', if the [[closure ( | Let ''X'' be a [[topological space]]. A subset <math>\scriptstyle A \subset X</math> is said to be '''dense''' in ''X'', or to be a dense set in ''X'', if the [[closure (topology)|closure]] of ''A'' coincides with ''X'' (that is, if <math>\scriptstyle \overline{A}=X</math>); equivalently, the only [[closed set]] in ''X'' containing ''A'' is ''X'' itself. | ||
==Examples== | ==Examples== | ||
# Consider the set of all [[rational number]]s <math>\scriptstyle \mathbb{Q}</math>. Then it can be shown that for an arbitrary [[real number]] ''a'' and desired accuracy <math>\scriptstyle \epsilon>0</math>, one can always find some rational number ''q'' such that <math>\scriptstyle |q-a|<\epsilon</math>. Hence the set of rational numbers are dense in the set of real numbers (<math>\scriptstyle \overline{\mathbb{Q}}=\mathbb{R}</math>) | # Consider the set of all [[rational number]]s <math>\scriptstyle \mathbb{Q}</math>. Then it can be shown that for an arbitrary [[real number]] ''a'' and desired accuracy <math>\scriptstyle \epsilon>0</math>, one can always find some rational number ''q'' such that <math>\scriptstyle |q-a|<\epsilon</math>. Hence the set of rational numbers are dense in the set of real numbers (<math>\scriptstyle \overline{\mathbb{Q}}=\mathbb{R}</math>) | ||
# The set of algebraic [[polynomials]] can uniformly approximate any [[continuity#continuous_function|continuous function]] on a fixed interval [''a'',''b''] (with ''b''>''a'') up to arbitrary accuracy. This is a famous result in analysis known as [[Weierstrass' theorem]]. Thus the algebraic polynomials are dense in the [[space of continuous functions]] on the interval [''a'',''b''] (with respect to the uniform topology). | # The set of algebraic [[polynomials]] can uniformly approximate any [[continuity#continuous_function|continuous function]] on a fixed interval [''a'',''b''] (with ''b''>''a'') up to arbitrary accuracy. This is a famous result in analysis known as [[Weierstrass' theorem]]. Thus the algebraic polynomials are dense in the [[space of continuous functions]] on the interval [''a'',''b''] (with respect to the uniform topology). | ||
Revision as of 14:27, 6 January 2009
In mathematics, denseness is an abstract notion that captures the idea that elements of a set A can "approximate" any element of a larger set X, which contains A as a subset, up to arbitrary "accuracy" or "closeness".
Formal definition
Let X be a topological space. A subset is said to be dense in X, or to be a dense set in X, if the closure of A coincides with X (that is, if ); equivalently, the only closed set in X containing A is X itself.
Examples
- Consider the set of all rational numbers . Then it can be shown that for an arbitrary real number a and desired accuracy , one can always find some rational number q such that . Hence the set of rational numbers are dense in the set of real numbers ()
- The set of algebraic polynomials can uniformly approximate any continuous function on a fixed interval [a,b] (with b>a) up to arbitrary accuracy. This is a famous result in analysis known as Weierstrass' theorem. Thus the algebraic polynomials are dense in the space of continuous functions on the interval [a,b] (with respect to the uniform topology).
