802: Immersed Submanifold with Boundary of C∞ Manifold with Boundary

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definition of immersed submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ immersion.

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Target Context

• The reader will have a definition of immersed submanifold with boundary of $C^{\mathrm{\infty }}$ 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: Structured Description

Here is the rules of Structured Description.

Entities:
$M^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\ast M$: $\subseteq M^{\prime }$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
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Conditions:
$M$ has a not-necessarily-subspace topology
$\wedge$
$M$ has an atlas such that the inclusion, $\iota :M\to M^{\prime }$, is a $C^{\mathrm{\infty }}$ immersion
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2: Note

This definition does not mean that for each arbitrary subset, $M\subseteq M^{\prime }$, a topology and an atlas can be chosen to make $M$ a $C^{\mathrm{\infty }}$ manifold with boundary; it means that if a topology and an atlas can be chosen for an $M$, $M$ (with the topology and the atlas) is an immersed submanifold with boundary of $M^{\prime }$.

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References

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