# Tagged Questions

**133**

votes

**7**answers

7k views

### Two commuting mappings in the disk

Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in $R^n$) to itself. Does there always exist a point $x$ such that ...

**87**

votes

**3**answers

4k views

### Is R^3 the square of some topological space?

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X ...

**58**

votes

**4**answers

7k views

### Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...

**57**

votes

**5**answers

3k views

### Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...

**48**

votes

**11**answers

3k views

### How should one think about non-Hausdorff topologies?

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" ...

**43**

votes

**9**answers

9k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**41**

votes

**2**answers

3k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**40**

votes

**27**answers

4k views

### Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...

**40**

votes

**16**answers

6k views

### Atiyah-Singer index theorem

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of ...

**38**

votes

**4**answers

2k views

### How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...

**38**

votes

**4**answers

2k views

### Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...

**37**

votes

**2**answers

3k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

**36**

votes

**7**answers

3k views

### Is there an algebraic approach to metric spaces?

It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...

**36**

votes

**8**answers

4k views

### What is a continuous path?

I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're getting nervous ...

**33**

votes

**5**answers

4k views

### Is the boundary $\partial S$ analogous to a derivative?

Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...

**32**

votes

**3**answers

1k views

### Does Euclidean space have a compact factor?

Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point?
Bing's Dogbone space is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with ...

**32**

votes

**1**answer

634 views

### Homeomorphisms and disjoint unions

Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and ...

**32**

votes

**7**answers

3k views

### Non-homeomorphic spaces that have continuous bijections between them

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \mapsto Y$ and $g: Y \mapsto X$?

**32**

votes

**4**answers

2k views

### Which topological spaces admit a nonstandard metric?

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric.
That is, let us define that a topological ...

**31**

votes

**4**answers

2k views

### How far is Lindelöf from compactness?

A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...

**31**

votes

**1**answer

555 views

### A Topology such that the continuous functions are exactly the polynomials

(I originally asked this question on Math.SE, where it received a lot of attention, but no solution.)
Which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous ...

**30**

votes

**7**answers

3k views

### Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)

In this question, Harry Gindi states:
The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.
Moreover, in the answers, Pete L. ...

**29**

votes

**7**answers

4k 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 ...

**29**

votes

**3**answers

2k views

### Why do finite homotopy groups imply finite homology groups?

Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. ...

**29**

votes

**0**answers

3k views

### Grothendieck's manuscript on topology

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible ...

**28**

votes

**9**answers

1k views

### Notions of convergence not corresponding to topologies

This question concerns the ramifications of the following interesting problem that
appeared on Ed Nelson's final exam on Functional Analysis some years ago:
Exam question: Is there a metric on the ...

**28**

votes

**5**answers

1k views

### Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?

Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...

**27**

votes

**5**answers

2k views

### Does “compact iff projections are closed” require some form of choice?

There are many equivalent ways of defining the notion of compact space, but some require some kind of choice principle to prove their equivalence. For example, a classical result is that for $X$ to be ...

**27**

votes

**2**answers

651 views

### A property of the unit circle

Let $(X,d)$ be a compact connected metric space with the property that for any distinct points $a,b$, $X\backslash \lbrace a,b\rbrace$ is disconnected. Clearly the unit circle has this property. Is ...

**27**

votes

**1**answer

1k views

### Every real function has a dense set on which its restriction is continuous

The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me ...

**27**

votes

**3**answers

2k views

### Independent evidence for the classification of topological 4-manifolds?

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...

**27**

votes

**1**answer

2k views

### Is every connected scheme path connected?

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**26**

votes

**3**answers

979 views

### Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:-
Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$ be ...

**26**

votes

**3**answers

3k views

### When is a Homology Class Represented by a Submanifold? [duplicate]

Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in ...

**26**

votes

**1**answer

700 views

### How many polynomial Morse functions on the sphere?

Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function of degree $1$, you ...

**26**

votes

**1**answer

1k views

### Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently ...

**25**

votes

**14**answers

2k views

### What are interesting families of subsets of a given set?

Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...

**25**

votes

**8**answers

3k views

### What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...

**25**

votes

**3**answers

2k views

### Why are profinite topologies important?

I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...

**25**

votes

**3**answers

6k views

### Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...

**25**

votes

**4**answers

901 views

### Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?

In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the catgory Top topological ...

**25**

votes

**4**answers

2k views

### Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...

**24**

votes

**6**answers

1k views

### When factors may be cancelled in homeomorphic products?

It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, ...

**24**

votes

**2**answers

827 views

### Can non-homeomorphic spaces have homeomorphic squares?

I an wondering if there are non-homeomorphic spaces $X$ and $Y$ such that $X^2$ is homeomorphic to $Y^2$.

**24**

votes

**6**answers

2k views

### Applications of string topology structure

Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure ...

**24**

votes

**2**answers

2k views

### “Transitivity” of the Stone-Cech compactification

Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \backslash \mathbb{N}$ be two non-principal elements of this ...

**24**

votes

**5**answers

2k views

### The ants-on-a-ball problem

Suppose I put an ant in a tiny racecar on every face of a soccer ball. Each ant then drives around the edges of her face counterclockwise. The goal is to prove that two of the ants will eventually ...

**23**

votes

**4**answers

1k views

### In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?

I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems ...

**23**

votes

**4**answers

3k views

### Fundamental group as topological group

Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...