Question

SOLVED #11

I SEND 10 SO YOU CAN SEE WHAT THE LAST PROBLEM MENTIONED IS ABOUT

IU. Extra Credit: This problem is about a process called the one-point compactification of a non-compact space. This is sometimes called the Alexandroff one-point compactifica- tion. The general idea is that you can sometimes add a single point to a non-compact 2 space and get a space that is compact: a) On the last homework set, you explored the stereographic projection map f S2-NR2. Extend this map to an abstract map F:SR(oo) where we define F restricted to S- N to simply be f and we define F(N)o. Explair why F is a homeomorphism. (Here again I am looking more for well-explained good intuition here rather than a rigorous proof. The first thing for you to decide is what you think open sets in R2 u [oo) should look like.) b) We call R2Ufoo) the "one-point compactification of R2. Why is Rufool compact? 11. Extra Credit: Inspired by the last problem, explain what the one-point compactifica- tion of R is. Then explain what the compactification of R" is.

Answer 1

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