Reference no: EM132174081

Baire Category Theorem and its Consequences

Need to do that logically and mathematically

1 Introduction

One form of the Baire Category Theorem (BCT) for complete metric spaces, which can be found in Rudin's book [WR76], Exercise 22, Chap. 3 (including the dense part), given as extra credit already in HW 3, is the following:

Exercise 6 (#1) Prove on any metric space X (even when X isn't complete) that BCT 1, BCT 2, BCT 3 are equivalent statements, i.e., any one of the statement being true implies all the other are true.

Now we can give some important consequences of the BCT which you are asked to prove as the next set of extra credit exercises.

Exercise 8 (#2) Prove the previous theorem and use it to give another proof of the fact that R is an uncountable set.

Exercise 10 (#3) Prove the Uniform boundedness principle. Hint: Let A_N = {x:|f(x) ≤ N for all f ∈ F}].

Exercise 12 (#4) Prove the previous theorem.

[Hint: Let A_N (∈) = {x : d (f_n (x) , f_m (x)) ≤ ε for all m, n > N}].

Attachment:- Assignment.rar