# One-point compactification

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• Nov 28th 2009, 08:07 AM
claves
One-point compactification
(I know I'm posting a lot of questions, but I have an exam on Tuesday...)

Find (and draw) a subspace of R^3 that is homeomorphic to the one-point compactification of X when

X = (0,1) x (0,1) \ [0,1/2] x [0,1/2]
• Nov 28th 2009, 06:41 PM
aliceinwonderland
Quote:

Originally Posted by claves
(I know I'm posting a lot of questions, but I have an exam on Tuesday...)

Find (and draw) a subspace of R^3 that is homeomorphic to the one-point compactification of X when

X = (0,1) x (0,1) \ [0,1/2] x [0,1/2]

X is homeomorphic to $\mathbb{R}^2$. One-point compactification $\mathbb{R}_\infty^{2}$ of $\mathbb{R}^2$ is homeomorphic to two-dimensional sphere $S^2=\{x = (x_1, x_2, x_3) \in \mathbb{R}^3: ||x||=1\}$.