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242 votes
28 answers
33k views

Your favourite application of the Baire Category Theorem

151 votes
8 answers
20k views

Do most mathematicians know most topics in mathematics?

  • 3,569
81 votes
2 answers
3k views

Factorial of a matrix: what could be the use of it?

  • 943
50 votes
8 answers
4k views

Why does intersection always preserve "closed" structures?

  • 1,358
46 votes
5 answers
6k views

What is (a) geometry?

39 votes
1 answer
1k views

Does there exist a continuous function $f: \Bbb R\to \Bbb R$ that is rational at (Lebesgue) almost every irrational, and irrational at every rational?

  • 5,710
38 votes
4 answers
4k views

When does a space admit a flat metric?

  • 2,605
26 votes
1 answer
459 views

Strengthening the intermediate value theorem to an "intermediate component theorem"

21 votes
3 answers
2k views

How to appreciate Riemannian geometry

  • 2,160
18 votes
1 answer
502 views

What is the history of the semidirect product?

  • 62.8k
14 votes
1 answer
547 views

How to stay productive while you are studying math? [closed]

  • 2,545
13 votes
1 answer
288 views

How can a non-mathematician intuitively understand the importance of algebraic varieties?

  • 1,724
12 votes
1 answer
278 views

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

12 votes
1 answer
268 views

Seems to be the optimal coefficient, $k$ integral inequality $\int_{0}^{1}f^2(x)dx\le \frac{423405}{246064}\left(\int_{0}^{1}f(x)dx\right)^2$

11 votes
0 answers
982 views

Random matrices, eigenvalue distribution.

  • 24.4k
10 votes
2 answers
853 views

Intuitive meaning of transitive action

  • 6,072
9 votes
2 answers
782 views

Why did Hilbert prove the Nullstellensatz?

9 votes
3 answers
1k views

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

9 votes
1 answer
230 views

When does the limit of $a_n$ exist where $a_{n+1}:=a_n+\frac{a_n^2}{n^2}?$

  • 845
9 votes
1 answer
210 views

Find the function $f$ such that $f(x)=\frac{f(2x)}{x+1}$.

7 votes
1 answer
102 views

$f : \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that for every fixed $y\in\mathbb{R}$, $f(x + y)-f (x)$ is a polynomial in $x$ [closed]

7 votes
4 answers
1k views

the determinant function is an open function?

  • 1,800
6 votes
1 answer
1k views

Mean value theorem on Riemannian manifold?

  • 620
6 votes
2 answers
277 views

How to think intuitively about compact injections?

6 votes
3 answers
627 views

What are some fields that intersect topology and number theory? [closed]

5 votes
3 answers
416 views

Why don't we have many non euclidean geometries out there?

  • 3,247
5 votes
1 answer
696 views

How to prove the existence of partitions of unity for smooth manifolds with boundary?

  • 1,382
5 votes
1 answer
6k views

$F$, Indefinite integral $\implies F$ is Absolutely Continuous

  • 251
4 votes
2 answers
595 views

What is the significance of Group Automorphism?

4 votes
2 answers
143 views

Why $\sum_{n=1}^{\infty}\frac{n\left(\sin x\right)^{n}}{2+n^{2}}$ is not uniformly convergent on $[0,\;\frac{\pi}{2})$?

  • 159