Basis

Definition

Let $X$ be a topological space. A collection $\mathcal{B}$ of subsets of $X$ is called a basis for $X$ if

1. Every $B\in \mathcal{B}$ is open
2. Every open subset of $X$ is the union of elements in $\mathcal{B}$

Examples

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

2)

Example

The collection of all singletons is a basis for the discrete topology.

$U={\bigcup }_{y\in U}\{y\}$

Proposition: continuity

Let $X,Y$ be spaces with $\mathcal{B}$ a basis for $Y$. A function $f:X\to Y$ is continuous $⟺$ $f^{-1}(B)$ is open for each $B\in \mathcal{B}$

So we do not acquire to check that every subset is open, we only need to check the elements of the basis are open.

Proposition: generation of basis

$\mathcal{B}$ is a basis for a unique topology on $X$ if and only if:

• ${\bigcup }_{B\in \mathcal{B}}=X$
• If $B_1,B_2\in \mathcal{B}$ and $x\in B_1\cap B_2$, then there is $B_3\in \mathcal{B}$ such that $x\in B_3\subseteq B_1\cap B_2$

math-tools

Proposition: relation to every neighborhood

Let $\mathcal{B}$ be a basis of topological space $X$ $N$ is a neighborhood of $p$ $⟹\exists B\in \mathcal{B}(p\in B\subseteq N)$

Basis in Vector space

30.6 Definition: Span

If all vectors in the vector space can be expressed as a linear combination of a set of vectors in the vector space, this vector set span (or generate ) this vector space. If this is a minimal subset, it is a basis.

Basis

$B$ is a basis if $B$ is linearly independent and $\mathbf{Span}(B)=V$

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