n-Dimensional Manifolds
 Definition: A set $M$ with a maximal $m$-dimensional atlas $\mathcal A = \{ (U_{\alpha}, \varphi_{\alpha}) : \alpha \in \Gamma \}$ is called an $m$-Dimensional Manifold if: 1) $M$ can be covered by a union of charts in $\mathcal A$ (Countability Condition). 2) For every pair of distinct points $p, q \in M$, $p \neq q$ there exists \alpha, \beta \in \Gamma $]] such that$p \in U_{\alpha}$,$q \in U_{\beta}$, and$U_{\alpha} \cap U_{\beta} = \emptyset$(Hausdorff Condition). Some people do not require a manifold to satisfy the Hausdorff condition. Such manifolds are called Non-Hausdorff Manifolds. Condition 1 gives us a slight notion of compactness for a manifold, while condition 2 allows us to separate distinct points in a manifold by disjoint charts. These two conditions are very important in the study of manifolds. For example, any smooth surface$\mathcal S$in$\mathbb{R}^3$is a$2$-dimensional manifold. The Hausdorff condition is particularly easy to check. Let$p, q \in S$,$p \neq q$. Then since$\mathcal S$is embedded in$\mathbb{R}^3$and since$\mathbb{R}^3$has the usual Euclidean topology, we can let$\displaystyle{r = \frac{\| x - y \|}{2}}$. Then$B(x, r)$and$B(y, r)$are disjoint open neighbourhoods of$x$and$y$. Taking the intersection of these open balls with$\mathcal S$will give us regions of$\mathcal S$that can be regarded as charts in a maximal atlas$\mathcal A$of$\mathcal S$to which both are disjoint. We will begin to look at some more examples of$n\$-dimensional manifolds in the next section.