Connected Sets

Real Analysis · Axiom Academy

A set is connected when it can't be split into two separated pieces — and in , that means exactly the intervals. 1. A Separation: When a Gap Splits a Set Start with what it means to be disconnected . A set S is disconnected if it can be cut into two pieces that don't touch — formally, a separation where A and B are nonempty, disjoint, and each open in S . Take the relatively-open sets and . The cut at 1.5 lands inside the empty gap (1,2) , so A and B are disjoint, nonempty, open in S , and cover S . 2. Connected: No Separation Exists A set is connected when it is not disconnected — no matter where you try to cut, you cannot produce a valid separation. An interval like [a,b] is the model case: it has no gap to slip an open cut into. 3. The Theorem: Intervals Are Exactly the Connected Subsets of This is the headline fact for the real line. A subset of is connected if and only if it is an interval — a single point, [a,b] , (a,b) , a half-open interval, a ray, or all of . 4. Continuity Carries It Forward: the IVT Connectedness is preserved by continuous maps : if S is connected and f is continuous, then the image f(S) is connected. On this is the engine of the Intermediate Value Theorem . A connected interval — one unbroken piece on the x -axis. Also connected — a y -interval with no gap, so every height c between f(a) and f(b) is hit. You've seen connectedness from the gap that breaks a set to the continuity that carries it forward. Scroll up to revisit any step.

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