Linked Questions

1
vote
2answers
141 views

Are some applications of the theorem of Baire? [duplicate]

Good night, someone could help me with some interesting applications of this important theorem and also I could say how I draw a dense subset in the complete space used in the theorem. Thanks for your ...
3
votes
2answers
146 views

Applications of Baire's Threom [duplicate]

In a lecture on Baire's Theorem (for complete metric spaces), I gave, for a rather advanced undergraduate class in Real Analysis (covering the theory of metric spaces and elements of general topology),...
1
vote
0answers
86 views

Baire's theorem [duplicate]

I want to know interesting applications of Baire's Category Theorem. For example existence of no where differentiable function. Can any body tell me some similar applications?
55
votes
14answers
5k views

Unexpected use of topology in proofs

One day I was reading an article on the infinitude of prime numbers in the Proof Wiki. The article introduced a proof that used only topology to prove the infinitude of primes, and I found it very ...
51
votes
5answers
3k views

Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?

Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space? Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
17
votes
12answers
4k views

What is the point of countable vs. uncountable sets?

I understand how to use these concepts and how to prove certain sets are countable or uncountable. However I don't get the point of it. What difference does it make whether a set is countable? People ...
31
votes
6answers
9k views

Examples of uncountable sets with zero Lebesgue measure

I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set. Thanks.
47
votes
1answer
13k views

Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
213
votes
1answer
8k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
4
votes
5answers
4k views

Prove that the open interval $(0, 1)$ contains uncountably infinite numbers.

Prove that the open interval $(0, 1)$ contains uncountably infinite numbers. Apparently, there is a way to prove this proposition using Cantor's diagonalization argument. How does that work? How ...
5
votes
4answers
10k views

Function example? Continuous everywhere, differentiable nowhere [duplicate]

Possible Duplicate: Are Continuous Functions Always Differentiable? If such a function exists, can anyone give an example of a function $f(x) : \mathbb{R} \longrightarrow \mathbb{R}$ that is ...
5
votes
1answer
2k views

Banach space in functional analysis

Prove that a closed subspace of a Banach space is also a Banach space. Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
9
votes
1answer
1k views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some $p_0>1.$ ...
1
vote
2answers
1k views

Two problems: When a countinuous bijection is a homeomorphism? Possible cardinalities of Hamel bases? [closed]

Let $X$ and $Y$ be topological spaces and let $f : X\rightarrow Y$ be a continuous bijection. Under which of the following conditions will $f$ be a homeomorphism? (a) $X$ and $Y$ are complete metric ...
5
votes
1answer
806 views

Application of Baire category theorem in Moore plane

The proof that Moore plane is not normal I have read was using Cantor's nesting theorem. But I heard that it is also possible to use Baire category theorem to prove and I want to know how. So, as ...

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