# 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 $\gamma:I\rightarrow X$ 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 ProofThatAPathConnectedSpaceIsConnected 2013-03-22 12:46:30 2013-03-22 12:46:30 n3o (216) n3o (216) 6 n3o (216) Proof msc 54D05