Intereting Posts

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$
Examples of functions where $f(ab)=f(a)+f(b)$
From norm to scalar product
Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$.
If two measures agree on generating sets, do they agree on all measurable sets?
Laplace transform of $\sin(\sqrt{t})$
application of strong vs weak law of large numbers
Continuous uniform distribution over a circle with radius R
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
connected manifolds are path connected
What does “curly (curved) less than” sign $\succcurlyeq$ mean?
Dyson-expansion like multidimensional integral
Irreducible solvable equation of prime degree
Series – Apostol Calculus Vol I, Section 10.20 #24
Axiom of Choice, Continuity and Intermediate Value Theorem

Let $X$ and $Y$ is a uniform spaces. Let $f$ is a uniformly continuous surjective function $X\rightarrow Y$.

Conjecture: If $X$ is totally bounded then $Y$ is also totally bounded.

- Graph of a function homeomorphic to a space implies continuity of the map?
- are two metrics with same compact sets topologically equivalent?
- For continuous function $ f:\mathbb S^1 \to \mathbb R$ there exists uncountably many distinct points $x,y$ such that $f(x)=f(y)$
- Closed surjective map
- preservation of completeness under homeomorphism
- Can compact sets completey determine a topology?

- Intersection of compact and discrete subsets
- An open set in $\mathbb{R}$ is a union of balls of rational radius and rational center.
- Metrizability of a compact Hausdorff space whose diagonal is a zero set
- Product of totally disconnected space is totally disconnected?
- In an N-dimensional space filled with points, systematically find the closest point to a specified point
- Why this two spaces do not homeomorphic?
- Free filter on Wikipedia
- Parametrization of $n$-spheres
- Theorem of Arzelà-Ascoli
- Orbit space of a free, proper G-action principal bundle

It seems that the conjecture is well known and can be proved straightforwardly. Let $f:(X,{\cal E})\to (Y,\cal F)$ be a surjective uniformly continuous map between uniform spaces and the space $(X,\cal E)$ is totally bounded. Let $F\in\cal F$ be an arbitrary entourage. Since the map $f$ is uniformly continuous, there exists an entourage $E\in\cal E$ such that $E\subset (f\times f)^{-1}(F)$. Since the space $(X,\cal E)$ is totally bounded, there exists a finite subset $A$ of $X$ such that $E[A]=X$. We claim that $F[f(A)]=Y$. Indeed, let $y\in Y$ be an arbitrary point. Since the map $f$ is surjective, there exists a point $x\in X$ such that $f(x)=y$. Since $E[A]=X$, there exists a point $a\in A$ such that $(a,x)\in E$. Since $E\subset (f\times f)^{-1}(F)$, we see that $(f(a),y)=(f(a),f(x))\in F$. Therefore $y\in F[f(A)]$.

- Prove that finite dimensional $V$ is the direct sum of its generalized eigenspaces $V_\lambda$
- All Sylow $p$-subgroups of $GL_2(\mathbb F_p)$?
- $f$ is continuous, is $1/f$ continuous
- How to compute $\limsup$ and $\liminf ,\;$ as $x\to+\infty,\;$ of $\;\sin(x^2+x+1/2)\sin(x+1/2)$?
- Show that if $f^{-1}((\alpha, \infty))$ is open for any $\alpha \in \mathbb{R}$, then $f$ is lower-semicontinuous.
- What is operator calculus?
- Absolute continuous family of measures
- A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic
- Radius of a cyclic quadrilateral given diagonals
- Intuition behind the definition of Adjoint functors
- What are the rules for convergence for 2 series that are added/subtracted/multiplied/divided?
- Not a small, not a big set
- A generalization of Cauchy's condensation test
- Number of combinations and permutations of letters
- A question on numerical range