Standard Basis for the Real Line

Topology · Axiom Academy

Working with the collection of open intervals as a basis Let's work through three problems that demonstrate how the collection of open intervals forms a basis for the standard topology on ℝ, and how we can use this basis to verify that sets are open. Problem 1: Verifying the Basis Conditions Problem 2: Rational Endpoints Generate Standard Topology Problem 3: Proving a Set is Open Using the Basis A basis for a topology must satisfy two conditions: (1) every point is in some basis element, and (2) intersections can be "covered" by basis elements The collection of all open intervals (a, b) forms a basis for the standard topology on ℝ A smaller collection, using only rational endpoints, generates the same topology due to the density of ℚ in ℝ To prove a set is open using a basis, show that for each point in the set, there's a basis element containing that point and contained in the set Finite unions of basis elements are always open sets in the generated topology

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