crimboi[21] 模仿大赛
Rotman algebraic topology上的题, 个人认为出的非常好, 极大增进了读者对计算过程的理解.
题面
A group which is also a topology space is called a topological group if and are continuous. It's then easy to demonstrate that is homeomorphism and if then with quotient map is topological group. Now if in addition is discrete and closed, and is simply connected, show that by imitating the computation of

先想再看提示哦
提示
Recall how one computes . One firstly shows that for any compact convex and there exists a unique for some integer s.t. by utilizing the fact that is uniformly continuous and thus dividing into intervals s.t. on any interval , and notice that is a homeomorphism. Then for any , setting and we can routinely prove is a well-defined isomorphism. For this problem we basically replace the above with : we firstly prove the unique existence of s.t. by seeking a neighborhood of s.t. restriction of on which is homeomorphism. Then by same trick we divide interval into parts s.t. on each part we have ; one familiar with analysis should see that this is equivalent to prove is uniformly continuous when viewing as a uniform space. Remaining parts are trivial.

愿你无需看标答
标答
Note that is continuous, and is closed (Why it is without being ?), so we can construct s.t. . For uniformly continuous part just notice that is also continuous.
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作者:HDD
链接:https://blog.hellholestudios.top/archives/1577
来源:Hell Hole Studios Blog
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