User user7090

20 votes

2 answers

4k views

Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$

elementary-set-theory

asked May 8, 2014 at 16:19

14 votes

5 answers

2k views

For every rational number, does there exist a sequence of irrationals which converges to it?

real-analysis

asked Mar 11, 2014 at 18:46

10 votes

5 answers

5k views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

asked Mar 31, 2014 at 19:01

9 votes

2 answers

123 views

If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.

real-analysis inequality

asked Apr 21, 2015 at 1:10

9 votes

1 answer

1k views

Fibre of a sheaf of $\mathcal{O}_X$-modules.

algebraic-geometry

asked Apr 23, 2017 at 23:43

8 votes

1 answer

901 views

Geometric significance of completion of a ring

algebraic-geometry commutative-algebra

asked Jul 25, 2017 at 0:40

8 votes

2 answers

243 views

When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

algebraic-geometry schemes ringed-spaces

asked Apr 17, 2021 at 23:12

8 votes

1 answer

453 views

Sets of measure zero and the Lebesgue differentiation theorem.

real-analysis measure-theory lebesgue-measure

asked Nov 16, 2014 at 5:00

7 votes

2 answers

823 views

How does the failure of integral closure of a coordinate ring relate to the cusp/singularity we see in the corresponding variety?

algebraic-geometry

asked Nov 17, 2017 at 4:07

7 votes

1 answer

1k views

Example of two atlases on a manifold $M$ which give rise to different sets of smooth functions on $M$.

differential-geometry manifolds differential-topology smooth-manifolds

asked Sep 23, 2016 at 3:26

6 votes

1 answer

4k views

The Line with two origins

general-topology

asked Mar 12, 2014 at 18:10

6 votes

1 answer

7k views

Homeomorphism between punctured plane and cylinder [duplicate]

general-topology

asked Feb 9, 2014 at 19:20

5 votes

0 answers

189 views

Direct sum of cohomology rings. How to interpret?

algebraic-topology

asked Dec 21, 2015 at 0:28

5 votes

2 answers

2k views

Proof that an affine scheme is quasi compact [duplicate]

abstract-algebra algebraic-geometry

asked Jul 26, 2016 at 21:59

5 votes

1 answer

414 views

Showing that $X \times \mathbb{A}^1$ is regular in codimension one.

algebraic-geometry

asked Sep 8, 2016 at 23:12

5 votes

0 answers

351 views

Definition of Exterior Algebra of a Graded Module

commutative-algebra graded-rings graded-modules

asked Apr 29, 2017 at 20:26

5 votes

1 answer

455 views

Moduli of Riemann surfaces (genus g curves) is a variety.

algebraic-geometry

asked Feb 22, 2017 at 1:44

5 votes

1 answer

98 views

Definition of module of Kähler differentials for a DG algebra

commutative-algebra homological-algebra differential-graded-algebras

asked Aug 3, 2021 at 22:42

4 votes

2 answers

254 views

Geodesics in the upper half plane

hyperbolic-geometry hyperbolic-functions

asked Apr 4, 2021 at 15:36

4 votes

1 answer

156 views

$k$-rational points of the automorphism functor of a scheme

algebraic-geometry schemes algebraic-groups group-schemes

asked Apr 17, 2021 at 23:05

4 votes

1 answer

387 views

Hartshorne Corollary 9.4 precise meaning of $\mathcal{F}_y$ and $\mathcal{F} \otimes k(y)$

algebraic-geometry

asked Apr 23, 2017 at 23:04

4 votes

1 answer

325 views

How does the following define a hermitian structure?

complex-geometry

asked Jun 3, 2017 at 2:18

4 votes

1 answer

3k views

A definition of a very ample divisor

algebraic-geometry definition

asked Sep 29, 2017 at 18:32

4 votes

1 answer

285 views

Why are holomorphic vector bundles over $\mathbb{C}^n$ trivial?

complex-geometry holomorphic-bundles

asked Jul 22, 2019 at 0:18

4 votes

1 answer

532 views

Isomorphism classes of $\mathbb{Z}[i]$ modules.

abstract-algebra

asked Aug 21, 2015 at 23:18

4 votes

0 answers

62 views

For a $3$-by-$3$ matrix, find $A$, $B$ integer $3$-by-$3$ matrices with determinant $\pm 1$ such that $AMB$ is diagonal.

linear-algebra abstract-algebra

asked Aug 5, 2015 at 18:48

4 votes

3 answers

577 views

Finite Union of Compact Sets Clarification

real-analysis

asked Dec 30, 2013 at 1:14

3 votes

3 answers

3k views

Radius of Convergence for analytic functions

complex-analysis

asked Dec 11, 2013 at 21:31

3 votes

1 answer

1k views

Prove that the set of lines in the plane is equal in cardinality to $\mathbb{R}$

elementary-set-theory

asked Feb 11, 2014 at 16:02

3 votes

2 answers

8k views

Commutator subgroup of a dihedral group.

abstract-algebra

asked Dec 5, 2013 at 2:53

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255 Questions

Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$

For every rational number, does there exist a sequence of irrationals which converges to it?

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.

Fibre of a sheaf of $\mathcal{O}_X$-modules.

Geometric significance of completion of a ring

When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

Sets of measure zero and the Lebesgue differentiation theorem.

How does the failure of integral closure of a coordinate ring relate to the cusp/singularity we see in the corresponding variety?

Example of two atlases on a manifold $M$ which give rise to different sets of smooth functions on $M$.

The Line with two origins

Homeomorphism between punctured plane and cylinder [duplicate]

Direct sum of cohomology rings. How to interpret?

Proof that an affine scheme is quasi compact [duplicate]

Showing that $X \times \mathbb{A}^1$ is regular in codimension one.

Definition of Exterior Algebra of a Graded Module

Moduli of Riemann surfaces (genus g curves) is a variety.

Definition of module of Kähler differentials for a DG algebra

Geodesics in the upper half plane

$k$-rational points of the automorphism functor of a scheme

Hartshorne Corollary 9.4 precise meaning of $\mathcal{F}_y$ and $\mathcal{F} \otimes k(y)$

How does the following define a hermitian structure?

A definition of a very ample divisor

Why are holomorphic vector bundles over $\mathbb{C}^n$ trivial?

Isomorphism classes of $\mathbb{Z}[i]$ modules.

For a $3$-by-$3$ matrix, find $A$, $B$ integer $3$-by-$3$ matrices with determinant $\pm 1$ such that $AMB$ is diagonal.

Finite Union of Compact Sets Clarification

Radius of Convergence for analytic functions

Prove that the set of lines in the plane is equal in cardinality to $\mathbb{R}$

Commutator subgroup of a dihedral group.