Intereting Posts

On prime ideals in a polynomial ring over a PID (from Reid's _Commutative Algebra_)
Helly's selection theorem (For sequence of monotonic functions)
Euler-Maclaurin Summation Formula for Multiple Sums
vanishing ideal of product of two affine varieties
sum of all positive integers less than $n$ and relatively prime to $n$
Center of Heisenberg group- Dummit and Foote, pg 54, 2.2
Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
Do continuous linear functions between Banach spaces extend?
What is $3^{43} \bmod {33}$?
Find the smallest cosntant $k>0$ such that $\frac{ab}{a+b+2c} + \frac{bc}{b+c+2a} + \frac{ca}{c+a+2b} \leq k(a+b+c)$ for every $a,b,c>0$.
Line integral over ellipse in first quadrant
Blowing up a singular point on a curve reduces its singular multiplicity by at least one
What is this probability of a random power of 2 beginning with 1?
Why doesn't Zorn's lemma apply to $[0,1)$?
How to find the equation of the graph reflected about a line?

Lebesgue lemma states that for every open cover $\{U_\alpha\}_{\alpha\in A}$ of a compact metric space $(X,\rho)$ there exists a number $d>0$ such that

$$

\forall x\in X \quad \exists \alpha_x\in A \quad (r<d \Rightarrow B_r(x)\subset U_{\alpha_x}).

$$

I wonder if this property is characteristic for compact metric spaces, that is if for every non-compact metric space there exists an open cover without a Lebesgue number.

- an “alternate derivation” of Poisson summation formula and discrete Fourier transformation
- first countable $\Leftrightarrow$ compact and Hausdorff?
- Does There exist a continuous bijection $\mathbb{Q}\to \mathbb{Q}\times \mathbb{Q}$?
- Is there a set of all topological spaces?
- Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
- Banach Measures: total, finitely-additive, isometry invariant extensions of Lebesgue Measure
- Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
- Difference between two definitions of Manifold
- Automorphism group of a topological space
- Sequence of partial sums of e in Q is a Cauchy sequence.

Give $\Bbb N$ the discrete metric:

$$d(m,n)=\begin{cases}1,&\text{if }m\ne n\\0,&\text{if }m=n\end{cases}$$

Clearly this space is not compact, but any positive $d\le 1$ is a Lebesgue number for every open cover of it.

**Added:** Having given the matter a bit more thought, I can prove the following theorem. Say that a metric space $\langle X,d\rangle$ is *Lebesgue* if every open cover of it has a Lebesgue number.

Theorem:Let $\langle X,d\rangle$ be a metric space. If $X$ has a non-convergent Cauchy sequence or an infinite closed discrete set of non-isolated points, then $X$ is not Lebesgue. In particular, every Lebesgue space is complete, and every perfect Lebesgue space is compact.

**Proof:** Suppose first that $\sigma=\langle x_k:k\in\omega\rangle$ is a non-convergent Cauchy sequence in $X$. Let $\langle X^*,d^*\rangle$ be the usual metric completion of $\langle X,d\rangle$, and let $p\in X^*$ be the limit of $\sigma$ in $X^*$. Let $V_0=X^*\setminus B_{d^*}(p,2^{-1})$, and for $k>0$ let $V_k=B_{d^*}(p,2^{-k+1})\setminus \operatorname{cl}_{X^*}B_{d^*}(p,2^{-k-1})$. For $k\in\omega$ let $W_k=X\cap V_k$. Then $\mathscr{W}=\{W_k:k\in\omega\}$ is an open cover of $X$ with no Lebesgue number.

Now suppose that $\{x_k:k\in\omega\}$ is a closed discrete set of non-isolated points in $X$. There is a pairwise disjoint, closure-preserving collection $\{V_k:k\in\omega\}$ such that $x_k\in V_k$ for each $k\in\omega$, so there is a sequence $\langle r_k:k\in\omega\rangle$ of positive real numbers such that $B_d(x_k,r_k)\subseteq V_k$ for each $k\in\omega$, and $\langle r_k:k\in\omega\rangle\to 0$. Let $$W=X\setminus\bigcup_{k\in\omega}\operatorname{cl}_X B_d\left(x_k,\frac{r_k}2\right)\;,$$ and let $\mathscr{W}=\{W\}\cup\{B_d(x_k,r_k):k\in\omega\}$; then $\mathscr{W}$ is an open cover of $X$ with no Lebesgue number. $\dashv$

The property: “every open cover has a lebesgue number” is called Lebesgue property.

For a metric space $M$, having the Lebesgue property is equivalent to:

For any metric spacr $M’$ every continuous function $f:M\rightarrow M’$ is uniforly continuous.

There are other properties equivallent to those, compactness is not one of them.

Here is a counterexample.

Consider the set IN of natural numbers with the the discrete topology, it’s not hard to see that IN is not compact and it has the Lebesgue property.

- On integrals related to $\int^{+\infty}_{-\infty} e^{-x^2} dx = \sqrt{\pi}$
- Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$.
- An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?
- Proof by the substitution method that if $T(n) = T(n – 1) + \Theta(n)$ then $T(n)=\Theta(n^2)$
- If $A$ is full column rank, then $A^TA$ is always invertible
- Is this a sound demonstration of Euler's identity?
- A basic question about exponentiation
- Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$
- A question on numerical range
- Computing the norm of a linear operator
- To show that $S^\perp + T^\perp$ is a subset of $(S \cap T)^\perp$
- Any manifold admits a morse function with one minimum and one maximum
- $T$ surjective iff $T^*$ injective in infinite-dimensional Hilbert space?
- Prove that $\,\sqrt n < 1 + \sqrt{\frac{2}{n}}\,$
- Let $B$ be a nilpotent $n\times n$ matrix with complex entries let $A = B-I$ then find $\det(A)$