Intereting Posts

$\mathbb{Q}/\mathbb{Z}$ has a unique subgroup of order $n$ for any positive integer $n$?
How to find the subgroups of S4 generated by these sets.
Radical of an ideal using Macaulay2 software.
Confused about why “disjointifying” implies “AC”
Integral ${\large\int}_0^\infty\frac{dx}{\sqrt{7+\cosh x}}$
Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.
Adjoint functors as “conceptual inverses”
Can vector spaces over different fields be isomorphic?
Long time existence of Ricci flow on compact surfaces of negative curvature
Supremum of all y-coordinates of the Mandelbrot set
Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices
Normalization of $k$
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 closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$

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?

- Two continuous functions that are the same in the rationals.
- Surprising applications of topology
- Homeomorphism between the Unit Disc and Complex Plane
- Two definitions of locally compact space
- A metric space in which every infinite set has a limit point is separable
- The cone of a topological space is contractible and simply connected

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?

- Urysohn's lemma with Lipschitz functions
- A union of balls centered at all the rationals on $$ with decreasing radius $r_n \to 0$ relation with $$
- Plane less a finite number of points is connected
- $G_{\delta}$ sets in locally compact Hausdorff or complete metric space are Baire spaces
- book with lot of examples on abstract algebra and topology
- Is there an example of a sigma algebra that is not a topology?
- How to prove that $\mathbb R^\omega$ with the box topology is completely regular
- Lang's treatment of product of Radon measures
- How to sort 4 (or more) values with non-piecewise functions?
- The empty set is a neighborhood?

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 $\Bbb N$.

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- Prove the Radical of an Ideal is an Ideal
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