Manifolds

---

An \(n\)-dimensional manifold is an object modeled locally on \(\mathbb R^n\). To make the intuitive notion a formal one, we define some notions. We say that two subsets \(U\) and \(V\) of Euclidean spaces are homeomorphic if there exists a bijection \(\varphi: U \to V\) such that both \(\varphi\) and its inverse are continuous. A subset \(M\) of some Euclidean space is said to be locally Euclidean of dimension \(n\) if every point of \(M\) has a neighborhood in \(M\) that is homeomorphic to a ball in \(\mathbb R^n\). The subset \(M\) is an \(n\)-dimensional manifold (but this is not the general definition of a manifold). For example, we consider a circle \(S^1\). For any point \(p\) of the circle, we have some small arc \(U\) containing \(p\) such that

\[U \cong (a,b) \in \mathbb R.\]

Thus circles are \(1\)-dimensional manifolds. In general, lines and curves are \(1\)-dimensional manifolds.

---