# 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 |
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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 |