Intereting Posts

Is it possible to alternate the law of mathematics?
Is this a correct use of the squeeze theorem?
The Affine Property of Connections on Vector Bundles
meaning of topology and topological space
Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?
Finding a pair of functions with properties
Show that $G$ is cyclic
Second order partial of $f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2}$
Application of Liouville's Theorem
What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?
Divisibility of multinomial by a prime number
Proof of rank-nullity via the first isomorphism theorem
Calculate the limit $\lim \limits_{n \to \infty} |\sin(\pi \sqrt{n^2+n+1})|$
transforming ordinary generating function into exponential generating function
Prove that the field F is a vector space over itself.

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.

- If locally convex topologies exhibit the same dual spaces, do they exhibit the same continuous linear operators?
- A continuous image of a second countable space which is not second countable.
- Why did mathematicians introduce the concept of uniform continuity?
- How is the epsilon-delta definition of continuity equivalent to the following statement?
- Axiomatizing topology through continuous maps
- What is a rigorous proof of the topological equivalence between a donut and a coffee mug?

- Triangulation of a simple polygon (elementary proof?)
- A simple curve of positive area
- Prove that there exists a Cauchy sequence, compact metric space, topology of pointwise convergence
- meaning of topology and topological space
- Prove Uncountable set minus a countable set is uncountable
- Torus cannot be embedded in $\mathbb R^2$
- To characterize uncountable sets on which there exists a metric which makes the space connected
- Classification of connected subsets of the real line (up to homeomorphism)
- Given any base for a second countable space, is every open set the countable union of basic open sets?
- Fundamental group of multiplicative group in Zariski topology

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 $y\in F[f(A)]$.

- What is the difference between Cartesian and Tensor product of two vector spaces
- Relation between root of a function and its derivative
- Continued fraction for some integrals by Ramanujan
- Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
- Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.
- Maximum principle of eigenvalue-like problem for harmonic equation
- Why do we treat differentials as infinitesimals, even when it's not rigorous
- The equation $x^3 + y^3 = z^3$ has no integer solutions – A short proof
- Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge?
- Families of functions closed under integration
- Finding constant to make integrals converge
- In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?
- What is the trellis diagram for a linear block code?
- How to show $e^{e^{e^{79}}}$ is not an integer
- Classification of general fibre bundles