Product Topology on Intervals

Topology · Axiom Academy

Worked examples exploring basis elements in the product topology on ℝ × ℝ Problem 1: Basis Elements for ℝ × ℝ Describe the basis elements for with the product topology. What geometric objects do they represent? Which of the following is a basis element for the product topology on ? Problem 2: Open Disk in Product Topology Show that the open disk is open in with the product topology. Problem 3: Half-Plane Openness Is the set open in ? Prove your answer using the basis criterion. What is the key insight in this proof? Basis elements for with product topology are open rectangles of the form . To prove a set is open, use the basis criterion : show every point has a basis element around it contained in the set. The open disk is open because we can inscribe rectangles inside it around each point. The half-plane is open because we can find rectangles around each interior point that stay below the line . Visualization helps: drawing the rectangles inside the sets makes the proofs more intuitive.

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