*Homework 1.1: if f, g is smooth, g**of is also smooth,
provided that gf is well-defined.** *

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__Answer: __

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*We suppose f: _{}and g: _{}. We have the following two given condition: for each x*

*there is F defined on open X containing x such that*_{}

* 2. there is G defined on open Y containing f(x)
such that *_{}

*We can construct open set U and V such that F map U onto
V; and V lies completely inside Y. Therefore, GF defined on U provides a well-defined
map satisfying the definition of smooth map. The construction can be as follow:*

*Let **K be an open set lies completely in Y and
contains f(x) and define V=*_{}. Let U=_{}. QED.

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*Homework 1.2: Give an example of differomorphism and a
proof.** *

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__Answer:__

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*The simplest example is the
identity map. A homeomorphism is a continuous mapping between two spaces that
has an inverse which is also continuous. Identity map does satisfy this
definition. It and its inverse are also smooth, obviously.
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