Basis

Let $(X, \tau)$ be a topological space.

Definition. A collection $\mathcal{B} \subseteq \tau$ is a basis for $\tau$ if every open set $U \in \tau$ is expressible as a union of members of $\mathcal{B}$: for every $U \in \tau$ and every $x \in U$, there exists $B \in \mathcal{B}$ with $x \in B \subseteq U$.

Proposition (Basis recognition). A collection $\mathcal{B}$ of subsets of a set $X$ is a basis for some topology on $X$ if and only if:

1. $\bigcup \mathcal{B} = X$ (every point of $X$ belongs to some member of $\mathcal{B}$), and
2. for every $B_1, B_2 \in \mathcal{B}$ and every $x \in B_1 \cap B_2$, there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$.

The topology $\tau_{\mathcal{B}}$ generated by $\mathcal{B}$ is the collection of all unions of members of $\mathcal{B}$.

Proof sketch. If $\mathcal{B}$ satisfies (1) and (2), define $\tau_{\mathcal{B}} = \{ \bigcup \mathcal{A} : \mathcal{A} \subseteq \mathcal{B} \}$. Condition (1) gives $X \in \tau_{\mathcal{B}}$; arbitrary unions of unions are unions; condition (2) ensures finite intersections of members of $\tau_{\mathcal{B}}$ are again in $\tau_{\mathcal{B}}$. The converse is immediate. $\square$

Proposition. A topological space $(X, \tau)$ is second-countable if and only if $\tau$ admits a countable basis.

Examples.

1. Metric spaces. The open balls $\{B(x, \varepsilon) : x \in X,\, \varepsilon > 0\}$ form a basis for the metric topology.
2. $\mathbb{R}^n$. The open boxes $(a_1, b_1) \times \cdots \times (a_n, b_n)$ with $a_i, b_i \in \mathbb{Q}$ form a countable basis, so $\mathbb{R}^n$ is second-countable.
3. Discrete topology. The singletons $\{ \{x\} : x \in X \}$ form a basis.

Remark. The notion of basis transfers to the categorical setting: a basis for a Grothendieck topology on a category generates the same sheaf condition as the full topology, via sites and covers. See also sheaf.