📂Topology

What is a Manifold?

Definition ¹

A topological space $X$ is called a $n$-dimensional manifold when it satisfies the following three conditions:

• (i): It is second-countable.
• (ii): It is Hausdorff.
• (iii): Every point of $X$ has a neighborhood homeomorphic to an open set in $\mathbb{R}^{n}$.

A $n$-dimensional manifold $X$ is said to have a boundary when it has the following two types of points:

• (1) Interior points: Every neighborhood of $x \in X^{\circ}$ is homeomorphic to $\mathbb{R}^{n}$.
• (2) Boundary points: Every neighborhood of $x \in \partial X$ is homeomorphic to $U^{n} := \left\{ \mathbf{x} \ | \ \mathbf{x} \in (\mathbb{R}^{+})^{n} \right\}$.

Description

Condition (iii) and being locally Euclidean are equivalent. That is, a manifold is a topological space that locally resembles Euclidean space. In particular, a $1$-dimensional manifold is called a Curve, and a $2$-dimensional manifold is called a Surface.

In the example above, the first and second are $1$-dimensional manifolds, but the third is not a $1$-dimensional manifold because it has a twisted part.

In particular, the following holds true for a $n$-dimensional manifold $X$ with a boundary and a $m$-dimensional manifold $Y$ without a boundary. $$ \partial (X \times Y) = X \times \partial Y $$