> [!definition] > > Let $M$ be a [[Manifold|manifold]] of class $C^p$ ($0 \le p \le \infty$) modelled on [$\real^n$](Euclidean%20Space). If $M$ is also [[Hausdorff Space|Hausdorff]] and [[Second Countable|second countable]], then $M$ is a **n-manifold**, which grants it the following topological properties: > - [[Locally Compact Hausdorff Space|Locally Compact, Hausdorff]], [[Separable Topological Space|Separable]]. > - [[Locally Path-Connected]], with countably many [[Connected Component|connected components]]. > - [$\sigma$-compact](Sigma%20Compact), [[Paracompact]] > - Has a [[Partition of Unity|partition of unity]] made from [[Compactly Supported|compactly supported]] $C^p$ functions subordinate to any [[Open Cover|open cover]].