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Homework Statement
Show that the one-point compactification of N (the naturals) is homeomorphic with the subspace {0} U {1/n : n is in N} of R.
The Attempt at a Solution
If we show that N is homeomorphic with {1/n : n is in N}, then this homeomorphism extends to the one-point compactifications of these spaces.
First of all, I assume we have the discrete topology on N? And on {1/n : n is in N}, too, right? Since if {1/n : n is in N} is a subspace of R, then the open sets in the subspace topology are all the intersections of the open sets of R with {1/n : n is in N}, which form its power set.
Let f(n) = 1/n. Clearly this is a bijection from N to {1/n : n is in N}. Clearly it's a homeomorphism, since if U is open in N, f(U) is open in {1/n : n is in N}, and backwards.
Since {0} U {1/n : n is in N} is the one-point compactification of {0} U {1/n : n is in N}, it must be homeomorphic to the one-point compactification of N.