I'm reading a proof that the union of pairwise non-disjoint connected sets is connected. So we let $W_\alpha$ be the connected sets in question and $W = \bigcup_{\alpha}W_\alpha$. Now we suppose that $F$ is a non-empty clopen subset of $W$ (we want to show that $F = W$). So, for some $\alpha_0, F \cap W_{\alpha_0}$ is non-empty, and apparently itself clopen, from which the intersection must equal $W_{\alpha_0}$ and the proof follows shortly after. I, however, can't see how it follows that the intersection is clopen.
Why is the intersection of a clopen set and another set clopen in this proof?
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general-topology
1 Answers
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$F\cap W_{\alpha_0}$ may well not be clopen in the whole space, but it’s clopen in the subspace topology on $W_{\alpha_0}$. Since $W_{\alpha_0}$ is connected in its subspace topology, and $F\cap W_{\alpha_0}\ne\varnothing$, it follows that $F\cap W_{\alpha_0}=W_{\alpha_0}$.
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0Oooooh, sorry, I misread. Any subset is clopen in **its** subspace topology, but certainly not in the subspace topology of $W_{\alpha_0}$. I managed to confuse myself with subsets and subspaces. – 2012-10-20