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Concept
Dense set
Summary
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
Formally, is dense in if the smallest closed subset of containing is itself.
The of a topological space is the least cardinality of a dense subset of
A subset of a topological space is said to be a of if any of the following equivalent conditions are satisfied:
The smallest closed subset of containing is itself.
The closure of in is equal to That is,
The interior of the complement of is empty. That is,
Every point in either belongs to or is a limit point of
For every every neighborhood of intersects that is,
intersects every non-empty open subset of
and if is a basis of open sets for the topology on then this list can be extended to include:
For every every neighborhood of intersects
intersects every non-empty
An alternative definition of dense set in the case of metric spaces is the following. When the topology of is given by a metric, the closure of in is the union of and the set of all limits of sequences of elements in (its limit points),
Then is dense in if
If is a sequence of dense open sets in a complete metric space, then is also dense in This fact is one of the equivalent forms of the .
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. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality.
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