Manifolds with Boundary - Shastri Definition 3.2.1 and Remark 3.2.1 ...

Peter

Well-known member
MHB Site Helper
I am reading Anant R. Shastri's book: "Elements of Differential Topology" ... ...

I am currently focused on Chapter 3: Submanifolds of Euclidean Spaces ... ... and in particular Section 3.2: Manifolds with Boundary ...

I am having trouble understanding Shastri's Remark following Definition 3.2.1 ...

Definition 3.2.1 and the remark following read as follows:

General Request

I am somewhat confused by Shastri's argument that

1. If $$\displaystyle x \in \partial X$$ ... then for every chart $$\displaystyle \psi$$ of $$\displaystyle X$$ at $$\displaystyle x$$, it follows that $$\displaystyle \psi (x) \in \mathbb{R}^{ k-1 } \times 0$$

and

2. It follows that $$\displaystyle X \text { minus } \partial X = \{ x \in X \ : \ \phi (x) \in \text{ int } ( H^k) \text{ for some chart } \phi \}$$

If anyone can clarify why 1 and 2 follow ... using a better, simpler and/or more detailed argument than Shastri ... then I would be most grateful ...

Question

In Figure 1 below, I have tried to summarize what Shastri argues cannot happen ... that is set $$\displaystyle A$$ containing the point $$\displaystyle y$$ is not in $$\displaystyle \mathbb{R}^{ k-1 } \times 0$$ ... but why exactly can't this happen ... ?

Further ... why exactly is impossible for a neighborhood of $$\displaystyle z$$ in $$\displaystyle \mathbb{R}^k$$ to be contained in $$\displaystyle B \subset H^k$$ ... ... ?

Help will be much appreciated ...

Peter