connected space ConnectedSpace 2001-11-17 09:42:50 2006-08-10 13:02:42 Definition connected connected components disconnected connectedness \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A topological space $X$ is said to be \emph{connected} if there is no pair of nonempty subsets $U,V$ such that both $U$ and $V$ are open in $X$, $U \cap V=\emptyset$ and $U \cup V=X$. If $X$ is not connected, i.e. if there are sets $U$ and $V$ with the above properties, then we say that $X$ is \emph{disconnected}. Every topological space $X$ can be viewed as a collection of subspaces each of which are connected. These subspaces are called the \emph{connected components} of $X$. Slightly more rigorously, we define an equivalence relation $\sim$ on points in $X$ by declaring that $x\sim y$ if there is a connected subset $Y$ of $X$ such that $x$ and $y$ both lie in $Y$. Then a connected component of $X$ is defined to be an equivalence class under this relation.