The Real Line is Connected

Topology · Axiom Academy

EXAMPLE The Real Line is Connected Classic proof using the least upper bound property of ℝ A topological space is connected if it cannot be written as the union of two non-empty disjoint open sets. Equivalently, is connected if the only subsets that are both open and closed (clopen) are and itself. The real line with the standard topology is connected. Example 1: Understanding the Contradiction Setup Let's understand what we're assuming when we suppose ℝ is disconnected. Example 2: Constructing the Supremum Now we use the completeness of ℝ. This is where the proof truly leverages what makes ℝ special! We'll show that both and lead to contradictions! We proved ℝ is connected by contradiction , assuming it could be partitioned into two disjoint open sets A and B. The completeness axiom (least upper bound property) was essential: every non-empty bounded subset of ℝ has a supremum in ℝ. The supremum s of A must lie in either A or B, but the openness of both sets leads to contradictions in each case. This proof technique shows why ℚ is not connected (it lacks the completeness property) but ℝ is connected . Connectedness of ℝ implies the Intermediate Value Theorem and many other fundamental results in analysis!

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