proof that a path connected space is connected


Let X be a path connected topological spaceMathworldPlanetmath. Suppose that X=AB, where A and B are non empty, disjoint, open sets. Let aA, bB, and let γ:IX denote a path from a to b.

We have I=γ-1(A)γ-1(B), where γ-1(A),γ-1(B) are non empty, open and disjoint. Since I is connected, this is a contradictionMathworldPlanetmathPlanetmath, which concludes the proof.

Title proof that a path connected space is connected
Canonical name ProofThatAPathConnectedSpaceIsConnected
Date of creation 2013-03-22 12:46:30
Last modified on 2013-03-22 12:46:30
Owner n3o (216)
Last modified by n3o (216)
Numerical id 6
Author n3o (216)
Entry type Proof
Classification msc 54D05