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Let $\overline{X} \rightarrow X$ be a covering space, where $\overline{X}$ is a Hausdorff space. Does $X$ have to be Hausdorff?

I assume there is a counterexample, but I couldn't find one yet.

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  • $\begingroup$ Try $X$ nonHausdorff and let the cover be a product with a discrete space. $\endgroup$
    – Randall
    Jul 14, 2018 at 21:00
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    $\begingroup$ Why is $\bar X$ Hausdorff then? $\endgroup$
    – Fan Zheng
    Jul 14, 2018 at 21:02
  • $\begingroup$ You’re right that doesn’t work. Never mind. $\endgroup$
    – Randall
    Jul 14, 2018 at 21:27
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    $\begingroup$ Example here. $\endgroup$
    – user574380
    Jul 14, 2018 at 21:33
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    $\begingroup$ It is true if the fibres $p^{-1}(x), x \in X$ are finite. See math.stackexchange.com/q/396989. $\endgroup$
    – Paul Frost
    Jul 14, 2018 at 21:39

1 Answer 1

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Here's a counterexample. Let $X=\mathbb{N}\cup\{\infty,\infty'\}$ where a set is open iff it is either contained in $\mathbb{N}$ or cofinite. Let $\overline{X}=X\times\mathbb{Z}$ with the following topology (not the product topology). A set $U\subseteq\overline{X}$ is open iff it satisfies the following two properties:

  • For all $n\in\mathbb{Z}$ such that $(\infty,n)\in U$, $(m,n+m)\in U$ for all but finitely many $m\in\mathbb{N}$.
  • For all $n\in\mathbb{Z}$ such that $(\infty',n)\in U$, $(m,n-m)\in U$ for all but finitely many $m\in\mathbb{N}$.

I now claim that the projection map $p:\overline{X}\to X$ is a covering map. First, it is easy to check that $p$ is continuous. Any $m\in\mathbb{N}$ has an evenly covered neighborhood, namely $\{m\}$ (since $p^{-1}(\{m\})=\{m\}\times\mathbb{Z}$ is discrete in $\overline{X}$). Finally, I claim $\infty$ and $\infty'$ have evenly covered neighborhoods. Indeed, if $V=X\setminus\{\infty'\}$, then $p^{-1}(V)$ is homeomorphic to $V\times\mathbb{Z}$ (with the product topology) via the map sending $(m,n)$ to $(m,n-m)$ and $(\infty,n)$ to $(\infty,n)$, so $V$ is an evenly covered neighborhood of $\infty$. Similarly, $W=X\setminus\{\infty\}$ is evenly covered since $p^{-1}(W)\cong W\times \mathbb{Z}$ by mapping $(m,n)$ to $(m,n+m)$.

Finally, $X$ is not Hausdorff ($\infty$ and $\infty'$ cannot be separated by open sets) but $\overline{X}$ is Hausdorff (we can get disjoint neighborhoods of $(\infty,n)$ and $(\infty',n')$ since the collections of points of the form $(m,n+m)$ and $(m,n'-m)$ only intersect at at most one point).

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