one point compactification of discrete space

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$X^*=X\cup \{\infty\}$ where the open neighbourhoods of $\infty$ are the cofinite sets.

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nomadicmathematician
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nomadicmathematician

Updated on August 01, 2022

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  • nomadicmathematician
    nomadicmathematician over 1 year

    Problem: What is the one point compactification $X^*$ of a discrete space $X$.

    In the case of $X$ being finite, $X$ itself is compact so the one point compactification would be merely $X$ $\bigcup$ {$\infty$}.

    Now in the case of $X$ being infinite I need to consider two cases.

    When $X$ is countably infinite, I have shown that $X^*$ is homeomorphic to {$0$} $\bigcup$ {$1/n$ | $n \in N$} in the subspace topology of the usual $R$.

    However, in the case of $X$ being uncountable, I cannot come up with any familiar space that $X^*$ is homeomorphic to. Can anyone help me out?

  • nomadicmathematician
    nomadicmathematician almost 9 years
    Thanks I was wondering if there is a homeomorphic space like in the case of countable discrete space but I guess there isn't one when the space is uncountable.