Basis for a topology

Definition

Let $X$ be a set. A basis for a topology on $X$ is a collection of subsets of $X$, $\mathcal{B}\subseteq\mathcal{P}(X)$ such that:

1. $\forall x\in X\exists B\in\mathcal{B}[x\in B]$ - every element of $X$ belongs to at least one basis element.
2. $\forall B_1,B_2\in\mathcal{B},x\in X\ \exists B_3\in\mathcal{B}[x\in B_1\cap B_2\implies(x\in B_3\wedge B_3\subseteq B_1\cap B_2)]$ - if any 2 basis elements have non empty intersection, there is a basis element within that intersection containing each point in it.

Note that:

• The elements of $\mathcal{B}$ are called basis elements

Topology generated by $\mathcal{B}$

If $\mathcal{B}$ is such a basis for $X$, we define the topology $\mathcal{J}$ generated by $\mathcal{B}$ as follows:

• A subset of $X$, $U\subseteq X$ is considered open (equivalently, $U\in\mathcal{J}$) if:
  • $\forall x\in U\exists B\in\mathcal{B}[x\in B\wedge B\subseteq U]$

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Claim: This $\mathcal{(J)}$ is indeed a topology

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TODO: Do this, see page 81 in Munkres - shouldn't be hard!

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See also

• Topological space
• Sub-basis for a topology
• Common topological constructs