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- About the proof of Lemma 1.35 - Smooth Manifold Chart Lemma in J. Lee book "Introduction to Smooth Manifolds"
I've a doubt regarding Lemma 1.35 (Smooth Manifold Chart Lemma) from J. Lee "Introduction to Smooth Manifolds"
The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are not disjoint though.
Lemma 1.35 (Smooth Manifold Chart Lemma)
Let ##M## be a set, and suppose we are given a collection ##\{U_{\alpha} \}## of subsets of ##M## together with maps ##\varphi_{\alpha}: U_{\alpha} \to \mathbb R^n## such that the following properties are satisfied:
(i) For each ##\alpha, \varphi_{\alpha}## is a bijection between ##U_{\alpha}## and an open subset ##\varphi_{\alpha}(U_{\alpha}) \subseteq \mathbb R^n##.
(ii) For each ##\alpha## and ##\beta##, the sets ##\varphi_{\alpha}(U_{\alpha} \cap U_{\beta})## and are open in ##\mathbb R^n##.
(iii) Whenever ##U_{\alpha} \cap U_{\beta} \neq 0##, the map ##\varphi_{\beta} \circ \varphi_{\alpha}^{-1}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta}) ## is smoot.
(iv) Countably many of the sets ##U_{\alpha}## cover ##M##.
(v) Whenever ##p,q## are distinct points in ##M##, either there exists some ##U_{\alpha}## containing both ##p## and ##q## or there exist disjoint sets ##U_{\alpha},U_{\beta}## with ##p \in U_{\alpha}## and ##q \in U_{\beta}##.
Then ##M## has a unique smooth manifold structure such that each ##(U_{\alpha}, \varphi_{\alpha})## is a smooth chart.
The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are not disjoint though.
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