The examples tag has no usage guidance.

**8**

votes

**3**answers

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### Example of a projective module which is not a direct sum of f.g. submodules?

This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...

**76**

votes

**6**answers

5k views

### Counterexamples in algebraic topology?

In this thread
Books you would like to read (if somebody would just write them...),
I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology".
My reason for doing so ...

**17**

votes

**6**answers

1k views

### Nonfree projective module over a regular UFD?

What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...

**13**

votes

**5**answers

3k views

### Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...

**5**

votes

**9**answers

2k views

### Examples of two different descriptions of a set that are not obviously equivalent?

I am teaching a course in enumerative combinatorics this semester and one of my students asked for deeper clarification regarding the difference between a "combinatorial" and a "bijective" proof. ...

**2**

votes

**2**answers

458 views

### Is there a non-trivial example for a 1-homogeneous function satisfying a specific inequality of second order?

Let $\mathbb{R}^n$ be the $n$-dimensional real vector space with Cartesian coordinates $x=(x^1,\ldots, x^n)\in \mathbb{R}^n$. I'm searching for a non-trivial example of a function $A:\mathbb{R}^n ...

**4**

votes

**3**answers

583 views

### Holomorphic function with a.e. vanishing radial boundary limits

Hello everybody.
I'm looking for an "easy" example of a (non-zero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1-} f(re^{i\phi})=0$.
...

**5**

votes

**2**answers

664 views

### Simple example of a sequence without computable modulus of convergence

Can anyone give a simple example of a sequence that converges, but there's no computable function that gives $N$ as a function of $\epsilon$, i.e., the modulus of convergence is not computable?
In ...

**21**

votes

**3**answers

2k views

### Clearing misconceptions: Defining “is a model of ZFC” in ZFC

There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with ...

**10**

votes

**1**answer

609 views

### Nondifferentiability set of an arbitrary real function

A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a continuous real function is that it is the union of ...

**4**

votes

**3**answers

1k views

### Nonessential use of large cardinals

In Awfully sophisticated proof for simple facts, we are asked for examples of complex proofs of simple results. To quote from the questioner's post, we are asked for proofs that are akin to "nuking ...

**15**

votes

**3**answers

2k views

### Non finitely-generated subalgebra of a finitely-generated algebra

Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...

**6**

votes

**2**answers

713 views

### Simple examples of equivariant homology and bordism

I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...

**32**

votes

**2**answers

2k views

### Difficult examples for Frankl's union-closed conjecture

Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...

**31**

votes

**14**answers

7k views

### Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition
that is derived from your everyday experience of physical reality. Your
intuitions about how the spin of a ball affects ...

**8**

votes

**3**answers

7k views

### Advanced Math Jokes [closed]

I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and ...

**13**

votes

**3**answers

1k views

### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that ...

**4**

votes

**4**answers

655 views

### An example of a non-paracompact tvs (over the reals, say)

What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...

**10**

votes

**3**answers

356 views

### Algebraic Curves and Phase Diagrams of Physical Systems

Lots of low degree curves arise naturally as the phase spaces of physical systems (that is, the curve parameterized by $(q,p)$ where $q$ is a generalized position variable and $p$ is a generalized ...

**16**

votes

**5**answers

5k views

### Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...

**30**

votes

**5**answers

2k views

### `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$.

[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold, say some space ...

**21**

votes

**2**answers

2k views

### Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...

**16**

votes

**3**answers

3k views

### Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to ...

**27**

votes

**11**answers

5k views

### Motivating the de Rham theorem

In grad school I learned the isomorphism between de Rham cohomology and singular cohomology from a course that used Warner's book Foundations of Differentiable Manifolds and Lie Groups. One thing ...

**4**

votes

**1**answer

703 views

### Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?

**1**

vote

**1**answer

1k views

### Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...

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votes

**19**answers

3k views

**36**

votes

**19**answers

7k views

### Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...

**32**

votes

**7**answers

7k views

### Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set).
Any countable set ...

**4**

votes

**1**answer

362 views

### Categories with products that preserve quotients

It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve ...

**9**

votes

**3**answers

1k views

### Concavity of $\det^{1/n}$ over $HPD_n$.

One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...

**143**

votes

**67**answers

47k views

### Awfully sophisticated proof for simple facts [closed]

It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...

**4**

votes

**0**answers

306 views

### Example of a Grothendieck pretopology satisfying a weak saturation condition

Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...

**14**

votes

**3**answers

849 views

### Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...

**5**

votes

**3**answers

457 views

### Nonmetrizable uniformities with metrizable topologies

I'm looking for such pathological examples of uniform spaces which are not metrizable, but whose underlying topology is metrizable. Willard in his General Topology text constructs such a uniformity ...

**1**

vote

**0**answers

669 views

### Example of smooth, proper but non-projective curve over an affine, connected base?

Would someone please give an example of a smooth, proper but non-projective curve $C/S$, where $S$ is affine and connected? I believe that whatever your example, $C/S$ must have genus $1$, admit no ...

**1**

vote

**2**answers

457 views

### Weakly initial sets - examples and nonexamples

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly ...

**5**

votes

**2**answers

676 views

### Canonical geometric examples

The proofs without words post has some great entries. I'm interested in a similar concept: examples where a problem in math or physics is accompanied by a geometric figure that illuminates some key ...

**2**

votes

**2**answers

257 views

### Is a compactly generated Hausdorff space functionally Hausdorff?

Question is the title. I suspect the answer is no, without some further conditions (clearly, normal is sufficient). Pointers to counterexamples would be appreciated, but not necessary.

**0**

votes

**1**answer

173 views

### Geometric explanation of an orbit space: Integer action on the affine line

Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space ...

**102**

votes

**59**answers

18k views

### Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following
passage in A Mathematician's Miscellany by J.E. Littlewood
(Methuen 1953, p. 79):
...

**1**

vote

**1**answer

1k views

### Ringed and locally ringed spaces

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...

**42**

votes

**7**answers

5k views

### Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...

**13**

votes

**17**answers

3k views

### What is your favorite isomorphism? [closed]

The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...

**3**

votes

**2**answers

517 views

### description of functions of conditionally negative type on a group

Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...

**5**

votes

**9**answers

1k views

### What category without initial object do you care about?

Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...

**31**

votes

**14**answers

6k views

### Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...

**4**

votes

**3**answers

512 views

### Natural examples of finite dimensional spaces with interesting 2-type

Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural ...

**7**

votes

**3**answers

821 views

### Is $\mathbb{A}²$ the universal smooth scheme which is a finite cover of $\mathbb{A}²/μ₂$?

One very handy (counter)example I often think about is the scheme $Spec(k[a,b,c]/(ab-c^2))$ (where $k$ is a field), which you may also know as $Spec(k[x^2,xy,y^2])$, as $\mathbb A^2/\mu_2$, or as the ...

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votes

**12**answers

2k views

### On proving that a certain set is not empty by proving that it is actually large

It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties.
For example, one can prove that ...