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 $\left\{{\left[\begin{array}{c}{x}_1,\dots ,x_n\end{array}\right]}^{\mathsf{T}}\in {\mathbb{R}}^n\mid x_n\ge 0\right\}$.

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

$$\mathrm{int}M$$

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 $M$:

$$\mathrm{int}M\cap \partial M=\varnothing \text{ and }\mathrm{int}M\cup \partial M=M$$

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

Charts

Definition: (Coordinate) Chart

Let $M$ be an $n$-dimensional topological manifold (with or without boundary).

A (coordinate) chart on $M$ is a pair $(U,\phi )$ of an open subset $U\subseteq M$ and a homeomorphism $\phi$ from $U$ to an open subset of ${\mathbb{R}}^n$ or ${\mathbb{R}}_{+}^n$. We call $U$ a coordinate domain and we call $\phi$ a coordinate map.

Definition: Coordinates

The component functions of $\phi$ are called (local) coordinates on $U$.

NOTATION

In such contexts, it is typical to denote the component functions of $\phi$ with superscripts: ${\phi }^1,\dots ,{\phi }^n$. We can then also write $(U,\phi )$ as $(U,{\phi }^1,\dots ,{\phi }^n)$.

Given a point $p\in M$, we call $p^1={\phi }^1(p),\dots ,p^n={\phi }^n(p)$ the coordinates of $p$ with respect to $\phi$.

Definition: Transition Map

Let $({\mathcal{D}}_{\phi },\phi )$ and $({\mathcal{D}}_{\psi })$ be charts on a manifold with or without boundary.

The transition map from $({\mathcal{D}}_{\phi },\phi )$ to $({\mathcal{D}}_{\psi },\psi )$ is the composition of $\psi$ with the inverse of $\phi$:

$$\psi \circ {\phi }^{-1}:\phi ({\mathcal{D}}_{\phi }\cap {\mathcal{D}}_{\psi })\to \psi ({\mathcal{D}}_{\phi }\cap {\mathcal{D}}_{\psi })$$

Definition: Atlas

Let $M$ be a manifold with or without boundary.

An atlas for $M$ is a collection of charts whose domains cover $M$.