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 ...
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),...
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?
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 ...
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 ...
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 ...
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.
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?
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 ...
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 ...
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 ...
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.
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 $p_0>1.$ ...
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 ...