483: Chart on Topological Manifold with Boundary

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A definition of chart on topological manifold with boundary

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Topics

About: topological manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological manifold with boundary.
• The reader knows a definition of homeomorphism.

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Target Context

• The reader will have a definition of chart on topological manifold with boundary.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

For any topological manifold with boundary, $M$, the pair of any open subset, $U\subseteq M$, and any homeomorphism, $\varphi :U\to \varphi (U)\subseteq {\mathbb{H}}^n\text{ or }{\mathbb{R}}^n$, where $\varphi (U)$ is any open subset of ${\mathbb{H}}^n\text{ or }{\mathbb{R}}^n$, denoted as $(U\subseteq M,\varphi )$

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2: Note

Logically speaking, $\varphi (U)$ can be allowed only an open subset of ${\mathbb{H}}^n$, because for any open subset of ${\mathbb{R}}^d$, we can instead take an open subset of ${\mathbb{H}}^d$ homeomorphic to it.

We allow also open subsets of ${\mathbb{R}}^d$ just for convenience: for a topological manifold (without boundary), an open subset of ${\mathbb{R}}^d$ is typically taken centered at the origin, and the argument would have to be changed (although it would be just a matter of translating the open subset into ${\mathbb{H}}^d$ or something) for a topological manifold with boundary if the open subset of ${\mathbb{R}}^d$ was not allowed: so, it is convenient for reusing arguments on topological manifold (without boundary) for topological manifold with boundary.

A topological manifold with boundary may be a topological manifold (without boundary), which is in fact a topological manifold with empty boundary, and in that case, each $\varphi (U)$ becomes a open subset of ${\mathbb{R}}^n$.

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References

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