First and second countability

Definition: first countable

First Countable

A collection ${\mathcal{B}}_p$ of neighborhoods of $p$ is called a neighborhood basis for $X$ at $p$ if every neighborhood of $p$ contains some $B\in {\mathcal{B}}_p$

note that this basis is a collection of neighborhoods

A space $X$ is called first countable if there is a countable neighborhood basis at each points.

Definition: second countable

Second Countable

A topological space $X$ 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 $\Vert x-p\Vert <\frac{q}{2}$

Relation between first and second countable

The first countable is a weaker condition comparing to the second countability

we can check: second countable $⟹$ first countable let $p\in X$ with $X$ is second countable. $X$ has countable basis $\mathcal{B}$. Consider ${\mathcal{B}}_p\subseteq \mathcal{B}$, ${\mathcal{B}}_p=\{B\in \mathcal{B}|p\in B\}$ $⟹|{\mathcal{B}}_p|\le |\mathcal{B}|⟹$ ${\mathcal{B}}_p$ is countable. By Proposition relation to every neighborhood we can easily prove this ${\mathcal{B}}_p$ is indeed a neighborhood basis.

Propositions

If $X$ is first countable, $A\subseteq X$ and $p\in X$

• $p\in \bar{A}⟺$ $p$ is a limit of a sequence in $A$.
• $p\in Int(A)⟺$ every sequence in $X$ converging to $p$ is eventually in $A$.
• $A$ closed $⟺$ $A$ contains every limit of every convergent sequence in $A$
  • note that ($A=\bar{A}⟺A$ is closed), so this can be shown by the first proposition
• $A$ open $⟺$ every sequence in $X$ converging to a point of $A$ is eventually in $A$.
  • use the second proposition. ($A=Int(A)$ $⟺$ $A$ is open) Explanations to notations: Definition basic concepts