diff --git a/spaces/S000135/properties/P000132.md b/spaces/S000135/properties/P000132.md index db67dd961..50ef1c38c 100644 --- a/spaces/S000135/properties/P000132.md +++ b/spaces/S000135/properties/P000132.md @@ -7,8 +7,8 @@ refs: 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