1
vote
1answer
147 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
1answer
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
3answers
370 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
0answers
137 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
2answers
334 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
1answer
144 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 ...
0
votes
0answers
85 views

Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...
4
votes
0answers
159 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
1answer
267 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
8answers
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
2answers
301 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
1answer
358 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 ...
0
votes
1answer
208 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
2answers
750 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
1answer
169 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
3answers
381 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
0answers
176 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
1answer
896 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
1answer
274 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
3answers
921 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
28answers
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
0answers
467 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
0answers
219 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
1answer
653 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
1answer
295 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
2answers
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
5answers
861 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
2answers
558 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
0answers
260 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
1answer
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
0answers
396 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
2answers
780 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
10answers
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
4answers
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
2answers
520 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
1answer
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
0answers
219 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
7answers
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. ...
3
votes
1answer
370 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
1answer
221 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 ...
25
votes
8answers
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
5answers
883 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
7answers
915 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
1answer
362 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 ...