Deprecated; please use a more specific tag.

**27**

votes

**9**answers

2k views

### Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...

**24**

votes

**2**answers

892 views

### Euler characteristic and universal cover

Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...

**21**

votes

**1**answer

2k views

### Math and Wormholes

Hopefully Math Overflow is the correct place for this. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...

**20**

votes

**1**answer

902 views

### Square roots of $\mathbb R^{2n}$

Recently, Richard Dore asked us if $\mathbb R^3$ is the cartersian square of some space, and Tyler Lawson answered beautifully in the negative.
The even powers of $\mathbb R$ were left out in that ...

**15**

votes

**1**answer

441 views

### Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.
Is $S\times S$ homeomorphic to $S$?
By Luzin ...

**12**

votes

**5**answers

1k views

### Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?

I'm curious about the following:
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...

**12**

votes

**3**answers

454 views

### Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following:
Given some simplicial complex $M$ with rational vertices representing a closed manifold,
can one decide (in finite time) if $M$ is ...

**10**

votes

**4**answers

747 views

### compact quotient

Let X be a topological space that is not too bad (let's say "not too bad" = "compactly generated Hausdorff"), and let ∼ be an equivalence relation such that X /∼ is compact Hausdorff.
Does there ...

**9**

votes

**4**answers

820 views

### When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...

**9**

votes

**2**answers

636 views

### Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...

**8**

votes

**0**answers

235 views

### Can a composition with itself of a universal self-map be non-universal?

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies.
DEFINITION A continuous map $u: ...

**8**

votes

**0**answers

462 views

### Characterization of Unusual Topologies of $\mathbb R$

Following some argument over a question on math.SE, I began to wonder:
We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...

**7**

votes

**3**answers

717 views

### What is the definition of continuity of set-valued functions?

According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W ...

**7**

votes

**2**answers

356 views

### The topology of open semi-algebraic sets (appl.: totally positive matrices)

Let $P_1,\ldots,P_r$ be polynomials over ${\mathbb R}^N$. I am interested in the homotopy class of the semi-algebraic set defined by
$$P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r.$$
Is there a ...

**6**

votes

**1**answer

619 views

### reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct:
Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...

**6**

votes

**4**answers

503 views

### On Pseudo-finite topological spaces

We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an ...

**6**

votes

**1**answer

221 views

### On the cardinality of perfect spaces with the countable chain condition

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces?
Recall that a ...

**5**

votes

**2**answers

788 views

### Two Definitions of “Character” of topological groups

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A character of $G$ is a ...

**5**

votes

**1**answer

199 views

### Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi_1$'s are trivial then ...

**5**

votes

**2**answers

887 views

### Construction of Serre Spectral Sequence

I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...

**5**

votes

**1**answer

265 views

### Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in
when mapping cone is contractible
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...

**5**

votes

**1**answer

430 views

### Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...

**5**

votes

**2**answers

340 views

### Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube

Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$?
Remarks and definitions:
1) The Hilbert cube $H$ is a compact metric space, where the metric is given by ...

**5**

votes

**4**answers

374 views

### degenerating immersion

Hi,
I would like to know if it exists a sequence of $C^2$ immersion $f_k: S^2 \rightarrow \mathbb{R}^3$ which converge (in C^2) to $z^2$ except on a finite set of point, i.e $f^k \rightarrow z^2$ in ...

**5**

votes

**0**answers

530 views

### Homotopy groups of a Bouquet of n-spheres

Let $$X=\vee_{\alpha\in A} S_{\alpha}^n$$ be a bouquet of $n$-spheres.
Q: How does one compute the homotopy groups $\pi_k(X)$?

**4**

votes

**3**answers

646 views

### When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say).
So far I had the following ...

**4**

votes

**1**answer

223 views

### Topological characterisation for a (closed irreducible) hyperbolic 3-manifold

Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. ...

**4**

votes

**3**answers

423 views

### Flat regions on surfaces of genus greater than 1

Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...

**4**

votes

**2**answers

468 views

### A question about some special compactifications of $\mathbb{R}$

We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a ...

**4**

votes

**2**answers

456 views

### Intersection forms of 4-manifolds with boundary

Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...

**4**

votes

**1**answer

290 views

### (Non)-exoticness of a diffeomorphism of a sphere

Suppose you have a standard sphere $S^n$ and a "standard" $S^{n-2}\subset S^n$. I am really thinking about $S^{n}\subset \mathbb{R}^{n+1}$ the usual sphere, and $S^{n-2}=S^n\cap \{x_0=x_1=0\}$. Let ...

**4**

votes

**2**answers

310 views

### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

**4**

votes

**0**answers

290 views

### Calculate the cohomology ring using Morse Theory [closed]

My question is: Is there any way to calculate the cohomology ring of a finite dimension manifold using the Morse theory?
The cohomology ring structure strongly rely on the following things
...

**3**

votes

**3**answers

468 views

### Connection between properties of Dynamical and Ergodic Systems

Hi All
While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...

**3**

votes

**3**answers

176 views

### Well-ordering with a topological property

Assuming the axiom of choice, is there a well-ordering of the reals such that every initial segment is closed (for the usual topology)? If the continuum hypothesis helps we can also assume it.
An ...

**3**

votes

**4**answers

367 views

### Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ...

**3**

votes

**1**answer

429 views

### The number of simply connected 4-dimension manifold

For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...

**3**

votes

**3**answers

286 views

### Embedding Theorem for topological spaces, and in general

There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...

**3**

votes

**1**answer

345 views

### A closed connected component in a topological space does not contain any path-connected subset?

Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected ...

**3**

votes

**1**answer

315 views

### Topological space with some conditions

Can one give an example of non-compact space $X$ which satisfies the following conditions:
the countable union of compact subsets is relatively compact,
for every closed noncompact subset $A$ of $X$ ...

**3**

votes

**1**answer

431 views

### Morse theory and adiabatic limits

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and ...

**3**

votes

**1**answer

334 views

### On One point Lindeloffication of topological spaces

As you Know when we define a topological space to be the one point compactification of the topological space $X$, we look for a compact space $Y$ such that $X\subset Y$ and $X$ is dense in $Y$ and ...

**3**

votes

**1**answer

332 views

### Determining continuous functions on Banach spaces

Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...

**3**

votes

**1**answer

135 views

### Examples of H-cogroups and a question about julia sets

The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441.
$\nabla \colon X\vee ...

**3**

votes

**0**answers

183 views

### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book, Lectures on tensor categories and modular functors by Bakalov ...

**3**

votes

**0**answers

135 views

### Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...

**2**

votes

**3**answers

351 views

### On the image of a G_\delta set under a continuous bijection

Let $X, Y$ be two metric spaces and $f$ be a continuous bijection (i.e. one-to-one map) from $X$ to $Y$. Let $E$ be a $G_{\delta}$ subset of $X$. I want to know weather the image $f(E)$ is also a ...

**2**

votes

**2**answers

555 views

### Are presheaves of constant functions sheaves?

Hey there, I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ ...

**2**

votes

**2**answers

320 views

### Defining a topology in the Power Set

I have the follwing question:
Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.
If the ...

**2**

votes

**1**answer

222 views

### Finding a good ordering of $\mathbb{Q}$

Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the ...