Countability Properties

A topological space is called second countable if it has a countable basis for its topology.

Example

Every Euclidean space is second countable. Recall this definition

1)

Example

Open balls form a basis of the topology on a metric space $M$.

defines open balls: $B_r(x)=\{y|d(x,y)<r\}$ defines the basis formed by open balls: $\mathcal{B}=\{B_r(x)|r\in {\mathbb{R}}_{>0},x\in M\}$ Proof: this is how a topology on a metric space is defined, see Metric Topology

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however, this basis is clearly not countable (real numbers are not countable)

Therefore, we define another basis: $\mathcal{B}=\{B_r(p)|r\in {\mathbb{Q}}_{>0},p\in {\mathbb{Q}}^n\}$ This is still a basis, and it’s countable Proof: Basis Consider $U\subseteq {\mathbb{R}}^n,x\in U$. There exists some $r\in {\mathbb{R}}_{>0}$ such that $B_r(x)\subseteq U$. There is $q\in {\mathbb{Q}}_{>0}$ with $q\le r$. By denseness there $p\in {\mathbb{Q}}^n$ such that $\Vert x-p\Vert <\frac{q}{2}$