# 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 ${\displaystyle [0,1]}$ is not used. As we said in class, if ${\displaystyle (f_{n})}$ is Cauchy in the uniform metric, then for any ${\displaystyle x}$, the sequence ${\displaystyle (f_{n}(x))}$ is Cauchy in ${\displaystyle {\mathbb {R} }}$, so it has a limit. Call that limit ${\displaystyle f(x)}$; it is not hard to show that ${\displaystyle f}$ is continuous and that ${\displaystyle f_{n}\to f}$. 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