The Basis Criterion

Topology · Axiom Academy

A practical way to verify that sets are open in a topology We've learned that a basis generates a topology by taking all possible unions of basis elements. But here's the practical problem: The Basis Criterion gives us a much more elegant answer. Instead of constructing explicit unions, we just need to check a simple local condition at each point. Let be a basis for a topology on a set . A set is open in if and only if: In other words: for each point in , we can find a basis element containing that point that stays entirely within . This is incredibly powerful because it reduces checking openness to a point-by-point verification. Step 3: Proving the "Only If" Direction Let's prove the easier direction first: If is open, then the criterion holds. By definition of basis-generated topology, we can write for some collection of basis elements . This is contained in (since it's part of the union). So we found a basis element containing that's contained in . ✓ This direction is straightforward: if is already a union of basis elements, then each point naturally lies in one of those basis elements! Step 4: Proving the "If" Direction Now the more interesting direction: If the criterion holds, then is open. Suppose for every , there exists with . For each , choose one such basis element and call it . Proof of ⊆: Each is contained in , so their union is too. Proof of ⊇: If , then by hypothesis there exists , so . Since equals a union of basis elements, it's open in . ✓

This is the written version of the interactive lesson above. See the full Topology course.