- What does it mean for a family of sets to form a basis for a topology?
Solution
1. A family \(\mathcal B\) of subsets of \(X\) is a basis if every point of \(X\) lies in some \(B\in\mathcal B\) and whenever \(x\in B_1\cap B_2\) with \(B_1,B_2\in\mathcal B\), there is \(B_3\in\mathcal B\) with \(x\in B_3\subseteq B_1\cap B_2\).
These two conditions are precisely the basis axioms.
2. The topology generated by \(\mathcal B\) consists of all unions of elements of \(\mathcal B\).
The basis axioms ensure these unions contain \(\varnothing,X\) and are closed under arbitrary unions and finite intersections.