I have a question about the definition of a manifold given in my analysis book. Here is the definition:(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex] 0 < k \le n [/itex]. A k-manifold in [itex] \mathbb{R}^n [/itex] of class [itex] C^r [/itex] is a set [itex] M \subset \mathbb{R}^n [/itex] having the following property: For each p in M, there is an open set V of M containing p, a set U that is open in either [itex] \mathbb{R}^k [/itex] or [itex] \mathbb{H}^k [/itex], and a continuous bijection [itex] \alpha: U \rightarrow V [/itex] such that:

1) [itex] \alpha [/itex] is of class [itex] C^r [/itex]

2) [itex] \alpha^{-1} [/itex] is continuous

3) [itex] D\alpha(x) [/itex] has rank k for each x in U

The map [itex] \alpha [/itex] is called a coordinate patch on M about p.

Note that the set [itex] \mathbb{H}^k [/itex] is upper half-space. That is, it is the set [itex] \{ x = (x_1, ..., x_k) \in \mathbb{R}^k : x_k \ge 0 \} [/itex]

My question is, why do require that the set U at least be open in [itex] \mathbb{H}^k [/itex]? What is so special about [itex] \mathbb{H}^k [/itex]?

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# Definition of Manifold

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