Basis (Topology)

Definition

A basis $\mathcal{B}$ is a collection of subsets of $X$ such that

• $\forall x\in X,\exists B\in \mathcal{B}\text{s.t.}x\in B\text{or}\bigcup _{B_i\in \mathcal{B}}{B}_i=X$
• $\forall B_1,B_2\in \mathcal{B},\forall x\in B_1\cap B_2,\exists B_3\text{s.t.}{B}_3\subset B_1\cap B_2$

$\mathcal{B}\subset \mathcal{T}\text{s.t.}(\forall U\in \mathcal{T},\exists \{B_i{\}}_{i\in I}\subset \mathcal{B}\text{s.t.}U={\bigcup }_{i\in I}{B}_i)$

Or, equivalently, a basis $\mathcal{B}$ for the Topology $\mathcal{T}$ of a Topological Space $(X,\mathcal{T})$ is a family of open subsets of $X$ such that every Open Set can be expressed as a union of subfamily of the family $\mathcal{B}$.

Examples

A Standard Topology on real numbers $(\mathbb{R},\mathcal{U})$ has a countable basis $\mathcal{B}=\{(a,b)\subset \mathbb{R}|a,b\in \mathbb{Q}\}$ A lower limit topology ${\mathcal{T}}_l$ generated by ${\mathcal{B}}_l=\{[a,b)\subset \mathbb{R}|a<b\}$ does not have a countable basis.

Facts

A basis can be constructed for a given Topology, where the basis is not unique. $\mathcal{B}\subset \mathcal{T}\text{s.t.}(\forall U\in \mathcal{T},\forall x\in U,\exists B\in \mathcal{B}\text{s.t.}x\in B\subset U)$

Suppose that a Topological Space $(X,\mathcal{T})$ has a countable basis $\mathcal{B}=\{B_n{\}}_{n\in \mathbb{N}}$. Then,

• Every open covering of $X$ contains a countable subcovering of $X$ (Lindelof Space)
• There exists a countable dense subset $D\text{s.t.}\bar{D}=X$ (Separable Space)

Consider a metrizable space $(X,\mathcal{T})$. Then, $X$ has a countable basis $\mathcal{B}=\{B_n{\}}_{n\in \mathbb{N}}$ if and only if $X$ is Lindelof Space if and only if $X$ is Separable Space

Every Compact metrizable space $(X,\mathcal{T})$ has a countable basis.

Consider a Topological Space $(X,\mathcal{T})$ has a countable basis $\mathcal{B}=\{B_n{\}}_{n\in \mathbb{N}}$, and a Subspace Topology $(Y,{\mathcal{T}}_Y)$ where $Y\subset X$. Then, The subspace topology also has a countable basis.

If a Topological Space $(X,\mathcal{T})$ has a countable basis $\mathcal{B}=\{B_n{\}}_{n\in \mathbb{N}}$, then any Subspace Topology $(Y,{\mathcal{T}}_Y)$ where $Y\subset X$, has a countable basis.