Intereting Posts

How to calculate the integral of $x^x$ between $0$ and $1$ using series?
Efficiently finding two squares which sum to a prime
Counting permutations of binary string where all ones within some distance?
Orthogonal Projection onto the $ {L}_{1} $ Unit Ball
Projection Matrix onto null space of a vector
are there $x_{1},x_{2} \in $ such that $x_{1}-x_{2}=1$ and $f(x_{1})=f(x_{2})$?
Operator for Laguerre polynomial
Positive operator is bounded
advantage of first-order logic over second-order logic
Is this a sound demonstration of Euler's identity?
When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?
The conjecture that no triangle has rational sides, medians and altitudes
Generating a regular sequence out of two
How Many Theorems (Tautologies) Exist of 5, 6, 7, 8, and 9 Letters?
Asymptotic behavior of Harmonic-like series $\sum_{n=1}^{k} \psi(n) \psi'(n) – (\ln k)^2/2$ as $k \to \infty$

I need to prove that an isometry $f$ on a compact metric space $X$ is necessarily bijective. I’ve got most of the proof, but I can’t figure out why any point in $X-f(X)$ would necessarily have to have some open neighborhood disjoint from $f(X)$.

- sequence lemma, relaxing some hypothesis of a theorem
- Lower Semicontinuity Concepts
- Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.
- Rudin's PMA Exercise 2.18 - Perfect Sets
- Can locally “a.e. constant” function on a connected subset $U$ of $\mathbb{R}^n$ be constant a.e. in $U$?
- Can $k$ be dense in $k$? where $p_xq_y-p_yq_x \in k^*$.
- Are there uncountably many non homeomorphic ways to topologize a countably infinite set?
- Error in book's definition of open sets in terms of neighborhoods?
- My proof of “the set of diagonalizable matrices is Zariski-dense in $M_n(\mathbb F)$”.
- Every neighborhood of identity in a topological group contains the product of a symmetric neighborhood of identity.

$f(X)$ is compact. If $x_0\notin f(X)$, because $X$ is separated for all $x\in f(X)$ exists two disjoints open sets $U_x$ and $V_x$ such that $x_0\in U_x$ and $x\in V_x$. We can find $n\in\mathbb N$ and $x_1,\cdots,x_n\in f(X)$ such that $f(X)\subset \bigcup_{j=1}^nV_{x_j}$. Now put $U:=\bigcap_{j=1}^nU_{x_j}$.

- Prove two parallel lines intersect at infinity in $\mathbb{RP}^3$
- Evaluating $\int \frac{\sec^2 x}{(\sec x + \tan x )^{{9}/{2}}}\,\mathrm dx$
- Finding points on the parabola at which normal line passes through it
- Why are vector valued functions 'well-defined' when multivalued functions aren't?
- Probability of $ \sum_{n=1}^\infty \frac{x_n}{2^n} \leq p$ for Bernoulli sequence
- help me understand derivatives and their purpose
- Is a perfect set a boundary?
- Continuous extension of a real function
- Calculating the Lie algebra of $SO(2,1)$
- Non-orthogonal projections summing to 1 in infinite-dimensional space
- Showing $\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}$
- Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?
- Could the concept of “finite free groups” be possible?
- How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
- Can all linear code be adapted so that the first bits of the encoded message are the bits from the original message?