1. The Baire category theorem - Aarhus Universitet

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Basic properties the notions of category (d) If h : S → S is a homeomorphism, E ⊆ S and h(E) are of the same category. It suffices to show that h(E) is of the first category if and only if E. Furthermore, since E = h−1 (h(E)), it suffice to show that h(E) is of the first category when E is. So assume that E = ∞ i=1 Ei, where each Ei has empty interior. Then h(E) = ∞ i=1 h (Ei) and h(Ei) = h Ei . If U ⊆ h Ei is open, h−1 (U) ⊆ Ei is also open. Klaus Thomsen 1. The Baire category theorem
Basic properties the notions of category (d) If h : S → S is a homeomorphism, E ⊆ S and h(E) are of the same category. It suffices to show that h(E) is of the first category if and only if E. Furthermore, since E = h−1 (h(E)), it suffice to show that h(E) is of the first category when E is. So assume that E = ∞ i=1 Ei, where each Ei has empty interior. Then h(E) = ∞ i=1 h (Ei) and h(Ei) = h Ei . If U ⊆ h Ei is open, h−1 (U) ⊆ Ei is also open. Since Ei has empty interior, this implies that h−1 (U) = ∅, i.e. U = h−1 (h(U)) = ∅. Klaus Thomsen 1. The Baire category theorem
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Proof of Baire’s category theorem
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1. The Baire category theorem - Aarhus Universitet

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