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P132 fix
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Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
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pzjp and prabau authored Jan 18, 2025
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name: Counterexamples in Topology
---

Since $X\setminus\{(0,0)\}$ is homeomorphic to the disjoint union of continuum many copies of the real line,
every open set in $X$ missing the origin is $F_\sigma$.
Since $X\setminus\{(0,0)\}$ is homeomorphic to the disjoint union of copies of the real line
and {S25|P132}, every open set in $X$ missing the origin is an $F_\sigma$.
It remains to show that the origin has a basis of open $F_\sigma$ neighborhoods.
Consider $U=\bigcup\{(-\varepsilon_\theta,\varepsilon_\theta)p_\theta: 0\leq \theta< \pi\}$, with $\varepsilon_\theta>0$
and $p_\theta=(\cos\theta,\sin\theta)$ for each angle $0\leq \theta < \pi$. Then the set
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