proof that a path connected space is connected

Let $X$ be a path connected topological space. Suppose that $X=A\cup B$, where $A$ and $B$ are non empty, disjoint, open sets. Let $a\in A$, $b\in B$, and let $\gamma :I\to X$ denote a path from $a$ to $b$.

We have $I={\gamma }^{-1}(A)\cup {\gamma }^{-1}(B)$, where ${\gamma }^{-1}(A),{\gamma }^{-1}(B)$ are non empty, open and disjoint. Since $I$ is connected, this is a contradiction, 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