By Marsden, Ratiu, Abraham.
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Extra info for Manifolds, tensor analysis, and applications
The notion of category can lead to interesting restrictions on a set. For example in a nondiscrete Hausdorﬀ space, any countable set is ﬁrst category since the one-point set is closed and nowhere dense. Hence in such a space every second category set is uncountable. In particular, nonﬁnite complete pseudometric and locally compact spaces are uncountable. 7-1. Let X be a Baire space. Show that (i) X is a second category set; (ii) if U ⊂ X is open, then U is Baire. 7-2. Let X be a topological space.
38 2. Banach Spaces and Diﬀerential Calculus If we set α = e2 , e2 , and β = − e1 , e2 , then this becomes 2 0 ≤ e2 , e2 2 2 e1 , e1 − 2 e2 , e2 | e1 , e2 | + | e1 , e2 | e2 , e2 , and so 2 2 e2 , e2 | e1 , e2 | ≤ e2 , e2 e1 , e1 . If e2 = 0, equality results in the statement of the proposition and there is nothing to prove. If e2 = 0, the term e2 , e2 in the preceding inequality can be cancelled since e2 , e2 = 0 by CI4. Taking square roots yields the statement of the proposition. Finally, equality results if and only if αe1 +βe2 = e2 , e2 e1 − e1 , e2 e2 = 0.
2 Proposition. Let X be a locally Baire space; that is, each point x ∈ X has a neighborhood U such that cl(U ) is a Baire space. Then X is a Baire space. Proof. Let A ⊂ X be residual, A = n≥1 On , where cl(On ) = X . Then if U is an open set for which cl(U ) is a Baire space, from the equality A ∩ cl(U ) = n≥1 (On ∩ cl(U )) and the density of On ∩ cl(U ) in cl(U ) (if u ∈ cl(U ) and u ∈ O, O open in X, then O ∩ U = ∅, and therefore O ∩ U ∩ On = ∅), it follows that A ∩ cl(U ) is residual in cl(U ) hence dense in cl(U ), that is, cl(A) ∩ cl(U ) = cl(U ) so that cl(U ) ⊂ cl(A).
Manifolds, tensor analysis, and applications by Marsden, Ratiu, Abraham.