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A manifold with a boundary is a topological space $(X, T)$ whose open sets have continuous one-to-one maps to open sets in half space. Half space is the region of $\mathbb{R}^{n}$ for which $x^{1} \geq 0$ (in Cartesian coordinates). The topology on half space is the induced topology from the standard one on $\mathbb{R}^{n}$.

Prove that the disk $\left\{(x, y)\mid x^{2}+y^{2} \leq 1\right\}$ is a manifold with boundary.

Anne Bauval
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Zeph Rodriquez
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  • What regularity do you want this manifold structure to have? – Sassatelli Giulio Dec 24 '22 at 20:00
  • Proposition 5.47 on page 121 of Lee's Introduction to Smooth Manifolds addresses precisely this type of question. For the function $f\colon \Bbb R^2 \to \Bbb R$ given by $f(x,y) = x^2+y^2$, we have that $1$ is a regular value and $${(x,y) \in \Bbb R^2 \mid x^2 + y^2 \leq 1} = f^{-1}\big((-\infty,1]\big),$$so it is a domain with regular boundary. The boundary is $C^\infty$ because so is the function $f$. – Ivo Terek Dec 25 '22 at 05:02

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