Intereting Posts

Tiling of regular polygon by rhombuses
Toss a fair die until the cumulative sum is a perfect square-Expected Value
Density of $\mathcal{C}_c(A\times B)$ in $L^p(A, L^q(B))$
Example of topological spaces with continuous bijections that are not homotopy equivalent
Universal property of initial topology
A question about the arctangent addition formula.
What is difference between cycle, path and circuit in Graph Theory
Solutions for diophantine equation $3^a+1=2^b$
Determine where a point lies in relation to a circle, is my answer right?
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
Book recommendation on plane Euclidean geometry
Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls
A family has three children. What is the probability that at least one of them is a boy?
Prove that if $p$,$q$ and $\sqrt{2}p+\sqrt{3}q$ are rational numbers then $p=q=0$
Resolvent of the Quintic…Functions of the roots

Let $X$ be a set and let $<_1,<_2$ be order relations on $X$.

Let $T_1,T_2$ be the topologies induced on $X$ respectively.

If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that $(X,<_1)$ and $(X,<_2)$ are order isomorphic?

- proving that $SO(n)$ is path connected
- Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.
- Do these $\sigma$-algebras on second countable spaces coincide?
- Axiomatizing topology through continuous maps
- A real function on a compact set is continuous if and only if its graph is compact
- Is the intersection of a closed set and a compact set always compact?

And a derived philisophical question: The other way around is easy to prove, so if this holds this means that, in some sense, homeomorphism between order topologies is equivalent to order isomorphism. What does that mean?

- Partitioning $\mathbb{R}^2$ into disjoint path-connected dense subsets
- Shrinking Group Actions
- Derived sets - prove $(A \cup B)' = A' \cup B'$
- Mapping homotopic to the identity map has a fixed point
- Is $2^\infty$ uncountable and is cardinality a continuous function?
- Image of open set through linear map
- Ring with spectrum homeomorphic to a given topological space
- Let $X$ be an infinite set with a topology $T$, such that every infinite subset of $X$ is closed. Prove that $T$ is the discrete topology.
- A continuous surjective function from $(0,1]$ onto $$
- Topology: reference for “Great Wheel of Compactness”

The answer is *no*. A trivial counterexample: if $\le$ is a linear order on $X$, $\langle X,\tau_\le\rangle$ and $\langle X,\tau_\ge\rangle$ are homeomorphic, but the orders are anti-isomorphic. Less trivially, the discrete topology on $X$ is generated by any discrete linear order on $X$, of which there are many. For example, if $\langle A,\le\rangle$ is any countably infinite linear order, the lexicographic order on $\langle A\times\Bbb Z\rangle$ induces the discrete topology on $A\times\Bbb Z$ and hence, via some bijection, on $\Bbb N$.

- the ratio of jacobi theta functions and a new conjectured q-continued fraction
- Probability and Laplace/Fourier transforms to solve limits/integrals from calculus.
- Is the embedding problem with a cyclic kernel always solvable?
- Weak topologies and weak convergence – Looking for feedbacks
- Determinant of circulant matrix
- Metric spaces problem
- A ‘strong’ form of the Fundamental Theorem of Algebra
- Random walk in the plane
- Evaluate $\sum_{k=1}^nk\cdot k!$
- Is there a “positive” definition for irrational numbers?
- Geometry Book Recommendation?
- How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
- Show that $(\mathbb{Z}/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
- Prove that there exists a Borel measurable function $h: \mathbb R \to \mathbb R$ such that $g= h\circ f$.
- A binary quadratic form and an ideal of an order of a quadratic number field