Generating a Topology from a Basis

Topology · Axiom Academy

LESSON Generating Topology from a Basis Building topological spaces from fundamental building blocks In topology, a basis is a collection of sets that serves as the building blocks for constructing a topology. Just as a vector space basis generates all vectors through linear combinations, a topological basis generates all open sets through unions. Let be a set. A collection of subsets of is called a basis if it satisfies: (B1) Covering: (every point is in some basis element) (B2) Intersection: If and , then there exists such that Let's examine a concrete example to see how basis conditions work in practice. (B1) Covering: Every point lies in the interval , so (B2) Intersection: If and , their intersection is either empty (which is fine) or another open interval , where and . For any point in this intersection, we can find a basis element (the intersection itself) containing it. Step 3: Defining Open Sets from a Basis Now we'll see how to construct a topology from a basis. The key idea is that open sets are built by taking arbitrary unions of basis elements. Let be a basis for a topology on . We define a set to be open if: In other words, is open if for every , there exists such that . Step 4: Proving This Gives a Valid Topology We must verify that the collection of open sets defined above satisfies the three topology axioms. Let be a basis on . Define to be the collection of all sets such that can be written as a union of elements from . Then is a topology on .

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