n-Dimensional Manifolds

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.

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