Intereting Posts

Lagrange Interpolating Polynomials – Error Bound
Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?
Showing that metric induces single unique topology on a finite set
Showing that $n$ is pseudoprime to the base $a$
Preserving the order of a sequence of real numbers
How to understand intuitively the Stolz-Cesaro Theorem for sequences?
What map of ring spectra corresponds to a product in cohomology, especially the $\cup$-product.
Why the terminology “monoid”?
Area and Polar Coordinates
Showing that a CCC with a zero object is the trivial category
Every sixth polynomial shares a factor of $(a^2-6)$
Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?
basic differential forms
Is $x^x=y$ solvable for $x$?
math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

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 $f(X)$.

- How to prove that a topological space is connected iff it has exactly two clopen subsets?
- Polish Spaces and the Hilbert Cube
- Prove that $\Bbb R^2 - \{0\}$ is homeomorphic to $S^1 \times \Bbb R$.
- Pullback of a covering map
- Generalization of “easy” 1-D proof of Brouwer fixed point theorem
- Understanding quotient topology
- Possibly not an acceptable proof for uncountablity of countable product of countable sets
- Is the Cantor set made of interval endpoints?
- Comparison of sequential compactness and limit point compactness.
- Can a continuous surjection from a Hilbert cube to a segment behave bad wrt Lebesgue measures?

$f(X)$ 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 $U:=\bigcap_{j=1}^nU_{x_j}$.

- A contradictory integral: $\int \sin x \cos x \,dx$
- How many different proofs can a theorem have?
- Can a set be neither open nor closed?
- What kinds of PDE can't be solved by separation of variables?
- Does a $\Pi_2^0$ sentence becomes equivalent to a $\Pi_1^0$ sentence after it has been proven?
- Inverse of Heine–Cantor theorem
- Differential Equations- Wronskian Fails?
- Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$
- Fraïssé limits and groups
- Is there a “counting groups/committees” proof for the identity $\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}$?
- Projective Normality
- How does linear algebra help with computer science
- Different definitions of trigonometric functions
- Why is the image of a C*-Algebra complete?
- trace of the matrix $I + M + M^2$ is