What is Density, topological form?

Example 12.17 (Density is relative)

“Dense” as defined here means dense in $\R$; a set can instead be dense in a part of the line only. The dyadics of $\intcc{0}{1}$, i.e. $D \cap \intcc{0}{1}$ (Exercise 10.8), meet every open interval included in $\intcc{0}{1}$ but of course miss $\intoo{2}{3}$ entirely: they are dense in $\intcc{0}{1}$, meaning $\overline{D \cap \intcc{0}{1}} = \intcc{0}{1}$. The general phrase “$A$ is dense in $B$” abbreviates $B \subseteq \overline A$ — always name the ambient set, since the weekend problem’s endpoints are dense in the Cantor set while being nowhere dense in $\R$: the same set, two truthful and opposite-sounding descriptions.

Example 12.18 (Handling density)

Three quick moves that recur constantly. Enlarging: if $A$ is dense and $A \subseteq B$, then $B$ is dense (every interval already meets $A$). Transporting: if $A$ is dense, so is $\lambda A + \mu$ for $\lambda \neq 0$ — an interval $I$ meets $\lambda A + \mu$ iff the interval $\frac{I - \mu}{\lambda}$ meets $A$; thus the odd multiples of $10^{-9}$, say, are dense. Intersecting fails: two dense sets can miss each other entirely ($\Q$ and $\R \setminus \Q$): density survives unions and affine maps, never intersections.

Example 12.12

$\overline{\intoo{0}{1}} = \intcc{0}{1}$; $\mathring{\intcc{0}{1}} = \intoo{0}{1}$; $\partial\intoo{0}{1} = \{0, 1\}$. For $A = \{\frac 1n : n \in \N^*\}$: $\overline A = A \cup \{0\}$, $\mathring A = \emptyset$, $\partial A = A \cup \{0\}$. For $\Q$: by density (Theorem 10.14) every real is adherent to $\Q$, so $\overline{\Q} = \R$ while $\mathring{\Q} = \emptyset$ (every interval contains irrationals): the boundary of $\Q$ is all of $\R$.