# Tagged Questions

**3**

votes

**1**answer

232 views

### Decomposition vs filtration vs stratification

Are there accepted/standard definitions of "decomposition", "filtration", and "stratification" of a topological space (or of a manifold, or of an algebraic variety) $X$?
I tend to understand ...

**10**

votes

**1**answer

642 views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...

**21**

votes

**2**answers

934 views

### A question about “Zariski dense” arguments

This question is a little basic, but I think it is consistent with the goals of MO.
My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...

**1**

vote

**1**answer

161 views

### Sober topological subspace

Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is ...

**1**

vote

**1**answer

116 views

### constructible set and fibre product

Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in ...

**6**

votes

**3**answers

404 views

### Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...

**6**

votes

**0**answers

166 views

### Zariski-homeomorphisms

This question is motivated by two questions at MO and
at MSE.
I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...

**3**

votes

**2**answers

337 views

### The quotient of $\mathbb{R}^{n}$ by a closed subset

Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of ...

**3**

votes

**1**answer

149 views

### Constructible subset of constructible set

Let $X$ be a topological space. Let $F \subset E \subset X$ be subsets. Assume that $E$ is constructible in $X$ and that $F$ is constructible in $E$. Is it true that $F$ is constructible in $X$?
We ...

**5**

votes

**0**answers

169 views

### Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...

**4**

votes

**1**answer

269 views

### Is every soft sheaf of countable $\mathbb Q$-vector spaces a direct sum of skyscraper sheaves?

Let $X$ be a finite-dimensional compact metrizable space (these properties might partially be irrelevant; on the other hand, the case $X=[0,1]$ is already interesting to me).
Let $\mathcal F$ be a ...

**20**

votes

**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

**5**

votes

**2**answers

318 views

### Given the vertices of a convex polytope, How can we construct its Half-Space representation

HI,
I have a question regarding convex polytopes. Let us say I have the vertices of a polytope which I name as $ V = \{v_1,\cdots,v_k\}$. Each of the $v_k$ are n-dimensional vectors, i.e. ...

**0**

votes

**1**answer

361 views

### A Question about SO(n)

My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...

**1**

vote

**1**answer

213 views

### Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition

Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...

**5**

votes

**2**answers

813 views

### Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.

Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
Condition 1: $X$ can be covered by ...

**1**

vote

**1**answer

170 views

### Ring of a Spectral Space

It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...

**2**

votes

**3**answers

382 views

### A simple question on the closure of the image of a morphism

Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a ...

**4**

votes

**0**answers

179 views

### Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...

**18**

votes

**1**answer

936 views

### Is $\mathbb{C}^2$ homeomorphic to $\mathbb{C}^2 - (0,0)$ with the Zariski topology?

A fellow grad student asked me this, I have been playing for a while but have not come up with anything. Note that $\mathbb{C}$ is homeomorphic to $\mathbb{C} - \{0\}$ in the Zariski topology - just ...

**3**

votes

**1**answer

279 views

### Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?

If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...

**6**

votes

**3**answers

985 views

### Is the preimage of the closure the closure of the preimage under a quotient map?

Let $f : X \to X/\sim$ be a quotient map from a topological space $X$ to the quotient space $X/\sim$ for $\sim$ some equivalence relation. Let $S \subseteq X/\sim$. Is it true that ...

**45**

votes

**28**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, ...

**1**

vote

**0**answers

485 views

### The “pullback presheaf” and the proper base change theorem in topology

Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
...

**1**

vote

**0**answers

220 views

### subset embedding gives trefoil knot [closed]

Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding ...

**1**

vote

**1**answer

667 views

### On Zariski Dense Subsets

Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of ...

**5**

votes

**1**answer

296 views

### Topological space associated to a real or complex scheme

Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...

**11**

votes

**2**answers

2k views

### Which platonic solids can form a topological torus?

8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...

**13**

votes

**5**answers

868 views

### What abstract nonsense is necessary to say the word “submersion”?

This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall ...

**2**

votes

**2**answers

560 views

### Varieties, Frechet Completions, and Regular Functions

Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...

**3**

votes

**0**answers

262 views

### Monomorphisms in geometry

What is known about monomorphisms in the following categories:
Schemes
Complex manifolds
$C^\infty$--manifolds
and any other kinds of geometric objects that you might think of.
How do we choose ...

**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 ...

**6**

votes

**0**answers

397 views

### Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...

**9**

votes

**2**answers

789 views

### Normal Varieties

Let X be a complex normal variety and U a subvariety that is open in the analytic topology. Then the map $\pi_1(U) \to \pi_1(X)$ coming from the map $U \subset V$ is surjective - why is this?
edited ...

**9**

votes

**10**answers

2k views

### Are nets and filters useful in geometry and topology?

Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...

**16**

votes

**4**answers

2k views

### Why are topological ideas so important in arithmetic?

For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...

**4**

votes

**2**answers

536 views

### Determining if two algebraic sets are homeomorphic

Is there an algorithm which, given two polynomials in $n$ variables with real coefficients, $p(x)$, and $q(x)$, will determine whether the zero sets $p^{-1}(0), q^{-1}(0)\subset R^n$, are homeomorphic ...

**2**

votes

**1**answer

414 views

### Sections of an etale space

In R.O.Wells book "Differential Analysis on Complex Manifolds" p. 44 proof of Theorem 2.2 part b) the author claims that any two sections of an etale space which agree at a point agree in some ...

**0**

votes

**0**answers

220 views

### Evenly submersive maps

A map $f: X\to S$ is called evenly submersive if each $s\in S$ has a neighborhood $W$ such that $p^{-1}W$ is covered by open sets $U\subset X$ diffeomorphic to $V\times W$ with $V$ open in $X_s$, and ...

**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. ...

**38**

votes

**7**answers

4k views

### “Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...

**3**

votes

**1**answer

371 views

### Sheaf condition and representability in the category Top

This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category ...

**3**

votes

**1**answer

225 views

### Is the coproduct of fibrant spectra fibrant again?

Define an $S^{1}$-spectrum $E$ to be a sequence of pointed simplicial sets $E_{n},\\ n=0,1,2...$ with assembly morphisms $\sigma_{n}:S^{1}\wedge E_{n}\rightarrow E_{n+1}$.
An $S^{1}$-spectrum $E$ is ...

**26**

votes

**8**answers

2k views

### When are there enough projective sheaves on a space X?

This question is being asked on behalf of a colleague of mine.
Let X be a topological space. It is well known that the abelian category of sheaves on X has enough injectives: that is, every sheaf ...

**7**

votes

**5**answers

893 views

### Analogues of the Weierstrass p function for higher genus compact Riemann surfaces

There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how ...

**12**

votes

**7**answers

938 views

### Can any topological space be the result of a scheme?

Maybe this is trivial but lets give it a try anyways..
Obviously there is a forgetful functor from schemes to topological space.. but is it surjective on objects? i.e. I ask whether any topological ...

**7**

votes

**1**answer

370 views

### Universal covers of domains in complex projective space

The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...