Bunifu UI WinForms V1.11.5.0
Bunifu UI WinForms V1.11.5.0
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DownloadQ:
Example of 2-to-1 morphism of locally ringed spaces
I would like to see an example of a 2-to-1 morphism of locally ringed spaces such that the image of the underlying topological spaces has an underlying topological space that is locally homeomorphic but not bijective.
I have in mind $\mathbb{C} \to \mathbb{C}^{*}$ with the affine line as topological space and the one-point compactification of the complement of the origin as a ringed space.
A:
Let $X$ and $Y$ be topological spaces, and consider the open immersion $U \to \mathbb{C}$ and its restrictions $U_{i} \to \mathbb{C}$ for $i \in \{0,1\}$. These maps are open immersions, and each pair $U_{i}, U_{j}$ have a common inverse, hence they are „almost bijective.”
At least one of these open immersions must be a homeomorphism. Call it $U \to Y$, so that $U \to Y$, $U_{0} \to Y$, and $U_{1} \to Y$ are open immersions that are pairwise almost bijective. Since they have at most two components, we can find a continuous section $Y \to U$. This means that $Y$ is homeomorphic to $U$, but $Y$ is not homeomorphic to $U_{i
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