Dense set

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

$𝐴$ is dense in $𝑋$ iff the smallest closed subset of $𝑋$ containing $𝐴$ is the whole of $𝑋$ .
$𝐴$ is dense in $𝑋$ iff the Closure of $𝐴$ in $𝑋$ is $𝑋$ itself, i.e. $C l 𝑋 (𝐴) = 𝑋$ .
$𝐴$ is dense in $𝑋$ iff the exterior of $𝐴$ is empty, i.e. $I n t (𝑋 ∖ 𝐴) = \emptyset$ .
$𝐴$ is dense in $𝑋$ with basis $B$ iff very basic neighbourhood $𝐵 \in B$ intersects with $𝐴$ so that $𝐴 \cap 𝐵 \neq \emptyset$ .

Proof of equivalence

#missing/proof

Examples

Metric topology

In a metric space $(𝑀, 𝑑)$ , $𝐴 \subseteq 𝑀$ is dense in $𝑀$ iff. every Open ball in $𝑀$ contains an element of $𝐴$ . This is the same as condition ^D4 above.

The set of rationals $ℚ$ is dense in the real numbers $ℝ$ 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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