# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### Explicit elements of $K((x))((y)) \setminus K((x,y))$

In an answer to the popular question on common false beliefs in mathematics Examples of common false beliefs in mathematics. I mentioned that many people conflate the two different kinds of formal ...
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### Examples of Super-polynomial time algorithmic/induction proofs?

In combinatorics, one can sometimes get an algorithmic proof, which loosely has the following form: -The proof moves through stages -An invariant is shown to hold by induction from previous stages -...
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### Comodule exercises desired

This Question is inspired by a Quote of Moore's "There are two ‘evil’ influences at work here: 1. we are toilet trained with algebras not coalgebras 2. some of us are addicted to manifolds and so ...
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### Can we have A={A} ?

Does there exist a set $A$ such that $A=\{A\}$ ? Edit(Peter LL): Such sets are called Quine atoms. Naive set theory By Paul Richard Halmos On page three, the same question is asked. Using the ...
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### Eta-products and modular elliptic curves

Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form  F=q\prod_{n>0}(1-q^n)^2(...
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### Vector spaces without natural bases

Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural". To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
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### Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...
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A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\... 13answers 2k views ### Applications of compactness Similar to this question: Applications of connectedness I want to collect applications of compactedness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ... 4answers 3k views ### What are your favorite finite non-commutative rings? When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand. There are many convenient examples of commutative rings, both finite and infinite, ... 13answers 2k views ### Applications of connectedness In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line). What are nice examples of applications of the idea ... 1answer 431 views ### Must finite groups with isomorphic commutators and quotients be isomorphic? Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ... 3answers 2k views ### The harmonic (series) beetle: live illustrations of mathematical theorems In my analysis class I use the following problem to illustrate the divergence of the harmonic series (consider this as a hint for solving it). Exercise. A beetle creeps along a 1-meter infinitely ... 1answer 401 views ### An example of a space which is locally relatively contractible but not contractible? A space$X$is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ... 6answers 3k views ### Fundamental group of the line with the double origin. In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole". This ... 10answers 3k views ### Examples of non-abelian groups arising in nature without any natural action It's said that most groups arise through their actions. For instance, Galois groups arise in Galois theory as automorphisms of field extensions. Linear groups arise as automorphisms of vector spaces, ... 2answers 598 views ### Gaining intuition for how submodules behave I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ... 9answers 698 views ### Non-real constants Constants are usually real numbers e.g. e, pi, gamma etc. Can you give examples of special constants that are not real? e.g. complex or p-adic constants. A real number in base10 can be viewed as the ... 1answer 2k views ### Sequence that converge if they have an accumulation point I am looking for classes of sequence, that converge iff they contain a converging sub-sequence. The basic example of such sequences are monotone sequences of real numbers. A more interesting ... 2answers 511 views ### What is an example of a non-regular, totally path-disconnected Hausdorff space? I need this for a counterexample: the multiplication in the fundamental group$\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from$\Omega \Sigma X_+$, fails to be continuous for ... 1answer 2k views ### Example for deterministic function with unbounded total variation and bounded quadratic variation It is well known that e.g.$sin(1/x)$is of unbounded total variation (in the interval [0,1] assuming$f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded quadratic variation. ... 3answers 2k views ### Algebras over the little disks operad Hello, The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied. My problem is the following: The "recognition principle" says that every "group-like" algebra over the ... 4answers 1k views ### Example of a smooth morphism where you can't lift a map from a nilpotent thickening? Definition. A locally finitely presented morphism of schemes$f\colon X\to Y$is smooth (resp. unramified, resp. étale) if for any affine scheme$T$, any closed subscheme$T_0$defined by a square ... 6answers 1k views ### What can you do with a compact moduli space? So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli ... 3answers 1k views ### Examples of Banach spaces and their duals There are many representation theorems which state that the dual space of a Banach space$X$has a particularly concrete form. For example, if$X = C([0,1],\mathbb R)$is the space of real-valued ... 0answers 233 views ### Krull rings and determinantal invariants During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ... 2answers 1k views ### Request: A Serre fibration that is not a Dold fibration A Serre fibration has the homotopy lifting property with respect to the maps$[0,1]^n \times \{0\} \to [0,1]^{n+1}$. A Dold fibration$E \to B$has the weak covering homotopy property: lifts with ... 4answers 5k views ### Good example of a non-continuous function all of whose partial derivatives exist What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous? 0answers 212 views ### Is the field of invariants$k(V)^G$purely transcendental over$k$? Reference: http://www.math.u-psud.fr/~colliot/mumbai04.pdf Proposition 4.3. on page 18 in the above reference reads as follows: Assume$k = \overline{k}$. If$V$is a finite dimensional vector space ... 12answers 3k views ### What are some fundamental “sources” for the appearance of pi in mathematics? I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ... 0answers 1k views ### Again about Bing's house with two rooms [duplicate] Possible Duplicate: How to show that the “bing’s house with two rooms” is contractible? I don't know why my question is closed? here, I make my question clearly, when "... 1answer 653 views ### Example of restriction of a finite morphism which is not finite Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties$X\...
Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective? Thanks! ...