1. The Baire category theorem - Aarhus Universitet

Proof of the corollary
Reminder ?: A subset E of S is dense when E ∩ U = ∅ for all open
and non-empty subsets U.
Examples: 1) Q is dense in R.
The polynomials are dense in C[0,1].
Nullermændene er tætte i min ældste søns værelse.
Proof of corollary using the theorem:
If S = ∞ i=1 Ei, where each Ei is nowhere dense, observe that
∅ = ∞ i=1 Ec
i ⊇ ∞ i=1 Ei
c c
and each Ei is dense and open. This
contradicts the theorem.
Klaus Thomsen 1. The Baire category theorem