Topological Manifolds#

Definition: (Topological) Manifold

An \(n\)-(topological) manifold is a second-countable Hausdorff space which is locally homeomorphic to a Euclidean space \(\mathbb{R}^n\).

Notation

\[ M^n \]

Theorem: Invariance of Dimension

A non-empty topological space cannot be both an \(n\)-manifold and an \(m\)-manifold with \(n \ne m\).

Proof

TODO

Definition: Dimension

We say that \(n\) is the dimension of \(M\).

Theorem: Open Subsets of \(n\)-Manifolds

Every open subset of an \(n\)-manifold is also an \(n\)-manifold.

Proof

TODO

Definition: Topological Manifolds with Boundary

An (topological) manifold with boundary is a second-countable Hausdorff space \((M, \tau)\) in which each point has a neighborhood which is homeomorphic to an open subset of the Euclidean space \(\mathbb{R}^n\) or to an open subset of the subspace \(\mathbb{R}^n_+\) of \(\mathbb{R}^n\) defined by \(\left\{\begin{bmatrix} x_1, \dotsc, x_n \end{bmatrix}^{\mathsf{T}} \in \mathbb{R}^n \mid x_n \ge 0\right\}\).

Definition: Dimension

We say that \(n\) is the dimension of \((M, \tau)\).

Definition: Interior

A point \(p \in M\) is an interior point of \((M, \tau)\) if it has a a neighborhood homeomorphic to an open subset of \(\mathbb{R}^n\).

The interior of \((M, \tau)\) is the set of all its interior points.

Notation

\[ \operatorname{int} M \]

Definition: Boundary

A point \(p \in M\) is a boundary point of \((M, \tau)\) if it has a a neighborhood homeomorphic to an open subset of \(\mathbb{R}^n_+\).

The boundary of \((M, \tau)\) is the set of all its boundary points.

Notation

\[ \partial M \]

Theorem: Interior is a Manifold

The interior of an \(n\)-dimensional manifold with boundary is an an \(n\)-dimensional manifold without boundary.

Proof

TODO

Theorem: Boundary is a Manifold

The boundary of an \(n\)-dimensional manifold with boundary is an an \((n-1)\)-dimensional manifold without boundary.

Proof

TODO

Theorem: Closedness of Boundary

The boundary of a manifold with boundary \(M\) is closed in \(M\).

Proof

TODO

Theorem: Disjointness of Interior and Boundary

The interior and boundary of a manifold with boundary \(M\) are disjoint whose union is \(M\):

\[ \mathop{\operatorname{int}} M \cap \partial M = \varnothing \qquad \text{ and } \qquad \mathop{\operatorname{int}} M \cup \partial M = M \]

Proof

TODO

Theorem: Manifold is a Manifold with Boundary

An \(n\)-dimensional manifold with boundary \(M\) is also an \(n\)-manifold without boundary if and only if \(\partial M = \varnothing\).

Proof

TODO