# 10-327/Classnotes for Monday December 6

See some blackboard shots at BBS/10_327-101206-142909.jpg.

Video:  Topology-101206

 Dror's notes above / Student's notes below
• Question. The fact that the metric space of real-valued functions on the unit interval with uniform metric is complete uses the fact that [0,1] is compact right? If the function space is defined on a non-compact topological space is that necessarily complete?... -Kai Xwbdsb 00:01, 20 December 2010 (EST)
• No, the compactness of [0,1] is not used. As we said in class, if (fn) is Cauchy in the uniform metric, then for any x, the sequence (fn(x)) is Cauchy in ${\mathbb R}$, so it has a limit. Call that limit f(x); it is not hard to show that f is continuous and that $f_n\to f$. Theorem 43.6 in Munkres is a slight generalization of this. Drorbn 07:12, 20 December 2010 (EST)

Thanks Dror.

Everybody good luck on the exam!-Kai

Great course! Thank you very much for all your help Dror and all the classmates in this class. -Kai