Density and Separability (稠密性と可分性) (Definition III.1.19)
- Density and Separability (稠密性と可分性) (Definition III.1.19) #Card
- $D \subset X$ is $\xi$-dense if for each $x \in X$ there exists a filter $\mathcal{F}$ with $D \in \mathcal{F}$ and $x \in \lim_\xi \mathcal{F}$. The density $d(\xi)$ is the least cardinality of a dense set; $\xi$ is separable if $d(\xi) \leq \aleph_0$.
Let $\xi$ be a convergence on $X$. A subset $D$ of $X$ is called $\xi$-dense if for each $x \in X$ there exists a filter $\mathcal{F}$ such that $D \in \mathcal{F}$ and $x \in \lim_\xi \mathcal{F}$. The density $d(\xi)$ of $\xi$ is the least cardinality of a $\xi$-dense set. A convergence is called separable if its density is countable.
- Each convergence on a countable set is separable.
- $A$ is $\iota_X$-dense $\iff A = X$；$A$ is $o_X$-dense $\iff A \neq \emptyset$.
- $\mathbb{Q}$ is $\nu_\mathbb{R}$-dense (Example III.1.20)；無理数の集合も $\nu_\mathbb{R}$-dense (Exercise III.7.4)。
位相の場合は Dense D / SeparableSpace X