Verifying Topology Axioms

Topology · Axiom Academy

EXAMPLE Verifying Topology Axioms Step-by-step verification of the three topology axioms This set should be in τ, but it's not! Always verify all three axioms systematically — even if some seem obvious, check each one carefully. Axiom 1 is usually the easiest to verify: just check that ∅ and X are explicitly in your collection. Axiom 2 (unions) often requires checking multiple cases or using set-theoretic arguments. Axiom 3 (intersections) applies only to finite intersections — infinite intersections need not be open. To show something is NOT a topology, you only need to find one axiom that fails. Visual diagrams help identify which sets are present and which are missing. For infinite topologies like the cofinite topology, use De Morgan's laws and properties of finite/infinite sets. Based on the example above, which approach is correct?

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