Intereting Posts

How to deal with this double summation?
Compute the derivative of the log of the determinant of A with respect to A
Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
Combinatorics, equality, $n$-permutations with $k$ cycles
Express Integer as Sum of Four Squares
What's the best way to measure mathematical ability?
The Parallelogram Identity for an inner product space
What books do you recommend on mathematics behind cryptography?
Proof $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$
Can every proof by contradiction also be shown without contradiction?
Find the total number of zeros in given range of decimal numbers
A multiple integral question
Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$
How can I understand and prove the “sum and difference formulas” in trigonometry?
Show that if $a$, $b$, and $c$ are integers such that $(a, b) = 1$, then there is an integer $n$ such that $(an + b, c) = 1$

The Wikipedia article on totally disconnected spaces seems to imply they are not necessarily Hausdorff (they are all $T_1$ though). What’s an example of a totally disconnected non $T_2$ space?

(A space is totally disconnected if all connected components are singletons.)

- Generalization of “easy” 1-D proof of Brouwer fixed point theorem
- Prove that the closure of complement, is the complement of the interior
- When is the union of topologies a topology?
- Path connectedness is a topological invariant?
- (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
- How to prove the Cone is contractible?

- Is it true that the unit ball is compact in a normed linear space iff the space is finite-dimensional?
- Cartesian product and closure
- How to prove every closed interval in R is compact?
- Intuition of the meaning of homology groups
- What is the opposite category of $\operatorname{Top}$?
- How many points does Stone-Čech compactification add?
- Whatever Happened to Nearness Spaces?
- Homeomorphisms between infinite-dimensional Banach spaces and their spheres
- partial converse of existence of covering spaces
- Prove that the identity map $(C,d_1) \rightarrow (C,d_\infty)$ is not continuous

Take $X = \mathbb N \cup \{ – \infty , + \infty \}$ with the topology where

- each point of $\mathbb N$ is isolated,
- each neighborhood of $- \infty$ and $+ \infty$ is a cofinite subset of $X$ (containing the respective point).

Note that for each $n \in \mathbb N$ the singleton $\{ n \}$ is clopen in $X$.

As $- \infty , + \infty$ cannot be separated by disjoint open sets, the space is not Hausdorff.

Suppose $A \subseteq X$ contains at least two points.

- If $A$ contains a point of $\mathbb N$, then for any $n \in A \cap \mathbb N$ the sets $\{ n \}$ and $X \setminus \{ n \}$ witness that $A$ is disconnected.
- Otherwise $A = \{ -\infty , + \infty \}$, in which case $X \setminus \{ + \infty \} , X \setminus \{ – \infty \}$ witness that $A$ is disconnected.

- Manipulating exponents of prime factorizations
- A problem on continuity of a function on irrationals for $f(x) = \sum_{r_n \leq x} 1/n^2$
- Is there a name for the group of complex matrices with unimodular determinant?
- Need help with understanding the Basis for a Topology
- Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n – 1}$
- What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?
- The alternating group is generated by three-cycles
- Unramification of a prime ideal in an order of a finite Galois extension of an algebraic number field
- Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$
- Double integral $\iint_D\arctan e^{xy}\,dy\,dx$
- Diagonalizable Matrices: How to determine?
- Connected Implies Path (Polygonally) Connected
- Gradient-descent algorithm always converges to the closest local optima?
- How many ways $12$ persons may be divided into three groups of $4$ persons each?
- QR factorization of a special structured matrix