Intereting Posts

How many combinations can I make?
Is that true that all the prime numbers are of the form $6m \pm 1$?
A good introductory discrete mathematics book.
Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$
a Circle perimeter as expression of $\pi$ Conflict?
Can monsters of real analysis be tamed in this way?
When can you simplify the modulus? ($10^{5^{102}} \text{ mod } 35$)
Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$
A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$
Elements in sigma algebra generated by sets (A,B)
Can we prove directly that $M_t$ is a martingale
A finite sum involving the binomial coefficients and the harmonic numbers
In a reduced ring the set of zero divisors equals the union of minimal prime ideals.
Solving Cubic Equations (With Origami)
A continuous function on $$ not of bounded variation

Assume

- $(X,\mathcal T)$ is a Tychonoff space.
- $(a_n)$ is a sequence in $X$.
- $a\in X$.
- for each continuous function $g:X\to [0,1]$,

$$g(a_n)\to g(a)$$

Is there an elementary proof for

$$a_n\to a$$

?

- $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?
- Distribution of zeroes of a continuous function
- Do simply connected open sets in $\Bbb R^2$ always have continuous boundaries?
- Principal ultrafilter and free filter
- Genus of a curve: topology vs algebraic geometry
- Is there “essentially only 1” Jordan arc in the plane?

- Uniqueness for a covering map lift: is locally connected necessary?
- Topology: reference for “Great Wheel of Compactness”
- Subnets and finer filters
- Separation axioms in uniform spaces
- What is the cardinality of the set of all topologies on $\mathbb{R}$?
- Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
- Which “limit of ultrafilter” functions induce a compact Hausdorff topological structure?
- every non-principal ultrafilter contains a cofinite filter.
- Natural derivation of the complex exponential function?
- Topology: Example of a compact set but its closure not compact

If $a_n$ doesn’t converge to $a$ then there exists an open neighbourhood $U$ of $a$ and a subsequence $b_n$ of $a_n$ such that $b_n$ lie outside $U$. Let $g:X\rightarrow \mathbb{R}$ be continuous such that $g(a)=0$ and $g(x)=1$ for all $x\notin U$ (here I am using the definition of Tychonoff space). Then $g(b_n)=1$ eventually and $g(a)=0$.

- How to find projection to polyhedron
- Divisibility Rule for 9
- How can I get the negation of $\exists!$ (unique existential quantification)?
- Integration using residues
- $R$ has a subring isomorphic to $R$.
- $\inf_{x\in}f(x)=\inf_{x\in\cap\mathbb{Q}}f(x)$ for a continuous function $f:\to\mathbb{R}$
- Does Euclidean geometry require a complete metric space?
- Summing $\frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1} + \cdots \text{ad inf}$
- How to prove inverse direction for correlation coefficient?
- Epimorphism and Monomorphism = Isomorphism?
- Finding all positive integers which satisfy $x^2-10y^2=1$
- Stable local minimizers
- Finitely generated field extensions
- Generating a random derangement
- How can I find equivalent Euler angles?