Basis for a topology

Definition

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

1. [ilmath]\forall x\in X\exists B\in\mathcal{B}[x\in B][/ilmath] - every element of [ilmath]X[/ilmath] belongs to at least one basis element.
2. [ilmath]\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)][/ilmath] - 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 [ilmath]\mathcal{B} [/ilmath] are called basis elements

Topology generated by [ilmath]\mathcal{B} [/ilmath]

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

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

Claim: This [ilmath]\mathcal{(J)} [/ilmath] 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