ProofThatAPathConnectedSpaceIsConnected 2002-06-10 06:52:27 2003-10-04 10:43:21 Proof path 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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.