Topology MOC

Dense set

A subset of a topological space is said to be dense iff every point of is either in or arbitrarily close to an element of . topology This is defined formally using the following equivalent conditions:

1. is dense in iff the smallest closed subset of containing is the whole of .
2. is dense in iff the Closure of in is itself, i.e. .
3. is dense in iff the exterior of is empty, i.e. .
4. is dense in with basis iff very basic neighbourhood intersects with so that .

Proof of equivalence

proof

Examples

Metric topology

In a metric space , is dense in iff. every Open ball in contains an element of . This is the same as condition ^D4 above.

The set of rationals is dense in the real numbers with the standard euclidean metric. A consequence of this is that a real number can be approximated to arbitrary precision by a rational number.

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