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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2015 • Question 1

Question

Math 6210 • Real Analysis Fall 2015 • Question 1

Prove the Baire category theorem. That is, show that if \((X,d)\) is a complete metric space and \(U_1,U_2,\ldots \subset X\) are open and dense, then \(\cap _{i=1}^{\infty }U_i\ne \emptyset \).

Raw LaTeX

Prove the Baire category theorem. That is, show that if \((X,d)\)
is a complete metric space and \(U_1,U_2,\ldots\subset X\) are open and dense,
then \(\cap_{i=1}^{\infty}U_i\ne\emptyset\).

Topic Tags

Baire Category and Uniform Boundedness Banach Spaces Compactness in Metric and Normed Spaces