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 Mentor P: 16,692 Yes, so you've proven that $$X=\{1/n~\vert~n\in \mathbb{N}\}$$ is noncompact, locally compact and Hausdorff. Thus the space has a one-point compactification. This is the space $$\{0\}\cup \{1/n~\vert~n\in \mathbb{N}\}$$ with the following topology $$\mathcal{T}=\{A~\vert~X\cap A~\text{open and}~(0\in A~\Rightarrow~X\setminus A~\text{compact})\}$$ This is, by definition, the topology of $$\{0\}\cup \{1/n~\vert~n\in \mathbb{N}\}$$. But this space already carries a topology (the Euclidian topology). So you still need to show that these two topologies coincide...