# Tagged Questions

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### Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
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### The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
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### explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
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### an equivalent statement of “morphisms of projective varieties are closed”

I am interested in seeing why the statement (1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed." implies the statement ...
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### Constructible sets

Is it possible to write down all the constructible sets in $\mathbf{C}$ (endowed with the Zariski topology) or some other "simple" space?
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### The closure of semialgebraic sets is semialgebraic.

I want to prove that the closure of semialgebraic subsets of $\mathbb{R}^n$ with respect to the Euclidean topology is semialgebraic. I may use the Tarski–Seidenberg theorem. Please give me not the ...
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### Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
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### How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
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### Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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### Topological Dimension via chains of connected nowhere dense closed sets

In algebraic geometry, one defines the dimension at a point of a variety $X$ as the length of the longest chain of irreducible closed subsets (in the Zariski topology of the variety) containing the ...
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### Homeomorphisms on Zariski topologies

I'm looking for a continuous bijection from a compact space to a non-Hausdorff topological space which isn't a homeomorphism. Since the identity $f:\mathbb{Z}\rightarrow\mathbb{Z},\ x\rightarrow x$ is ...
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### dimension of a subspace of a flag variety

Let $X$ be a topological space. If $X = \bigcup U_\alpha$ is an open covering of $X$ then $$\dim X = \sup_\alpha \dim U_\alpha.$$ Now suppose that $X = \coprod U_\alpha$, i.e., $X$ is the disjoint ...
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### The torus as a complex variety

I'm interested in the topological torus, ie. the homeomorphism class of $S^1\times S^1$. Clearly, it can be realized as the real algebraic variety in $\mathbb{R}^4$ as the solution set to ...
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### compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
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### Are the fibers of a flat map homotopy equivalent?

At the end of the Wikipedia article on Deformation Retract, there is the following sentence: Two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger ...
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### Any Real Algebraic Variety Has Finitely Many Path Components?

Consider the following statement: Statement: Any real algebraic variety has only finitely many path components. By 'path component' I mean: Let $X$ be a real algebraic variety. Then $X$ is a subset ...
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### Homeomorphism class of GL_n?

For example, it is easy to see that $GL_1(\mathbb{C})$ is a plane minus a point, and $GL_2(\mathbb{R})$ is $\mathbb{R}^4$ with a topological (half-open) cube removed (since the matrices of determinant ...
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### Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
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### intersection of two irreducible spaces not irreducible

Let $X$ be a topological space and let $Y_1,Y_2$ be two distinct irreducible subsets with none containing the other. Then it seems to me that $Y_1 \cap Y_2$ need not be irreducible. Is that right? ...
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### Zariski topology and polynomial maps

I've read on my book that Zariski topology is coarser than every topology in which polynomial maps are continous, but no proof of this fact is given. Could someone sketch me the proof of this?
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### The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
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### How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
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### $X$ being locally closed is not equivalent to every extension by $0$ of a sheaf $F$ being unique?

In Tennison's book "Sheaf Theory", the author presents a proof that there is a unique extension by $0$ for a sheaf $F$ over $X$ iff $X \subset Y$ is locally closed . However, apparently in the proof ...
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### What Information/Advantage do we Gain by Substituting a Continuous Map by a Fibration?

I'm trying to understand the usefulness of "substituting" a continuous map f , by a fibration F. By substituting, I mean there is the result that given a continuous map $f:X \rightarrow Y$ , for ...
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### Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff.

Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff. Here $Spec(R)$ means the Zariski topology over the set of all prime ideals of $R$. To put it ...
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### Irreducibility of set

Let $Y$ be a subset of the topological space $X$ and let $\{U_i\}$ be an open cover of $X$. 1.If $Y$ is not contained in any $U_i$ then $Y$ is irreducible? 2.If $\{V_j\}$ is an open cover of $Y$ ...