Connectedness and Path-Connectedness

GRE Math Subject · Axiom Academy

LESSON Connectedness & Path-Connectedness Connected spaces, path-connected spaces, components, and the IVT as a topological theorem Definition: A topological space is disconnected if there exist two nonempty, disjoint open sets with . Such a pair is called a separation of . A space is connected if no separation exists. Equivalent formulations (all say the same thing): is connected iff the only subsets that are both open and closed (clopen) are and is connected iff every continuous function is constant Any interval in (open, closed, half-open, or all of ) (the rationals) — for any irrational , the sets form a separation Any set with the discrete metric and more than one point Step 2: Properties of Connected Spaces Theorem: The continuous image of a connected space is connected. That is, if is continuous and is connected, then is connected. Intermediate Value Theorem (topological version): If is continuous and is connected, then is an interval. In particular, if is a connected subset of and with , then . If are connected subsets with , then is connected The closure of a connected set is connected Products of connected spaces are connected ( connected iff each is) Definition: A path in from to is a continuous function with and . A space is path-connected if for every pair of points , there exists a path from to . Theorem: Path-connected implies connected. The converse is false! This is one of the most important facts for the GRE.

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