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