# Tagged Questions

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### Are any (non-empty) Euclidean open sets dense in the Zariski topology?

It's well known and easy to show that every Zariski open set is dense in the Zariski topology. However I search the web and didn't find an answer to my question, which I believe is true. My ...
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### Image of Regular Map

Determine the image of the regular map $f: A^2 \to A^2$, $f(x,y)=(x,xy)$ and describe it from the point of view of topology. Would the image of f be $A^2$, because every point of $A^2$ is still in the ...
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### Product topology of Affine Varieties

I want to prove that $X \times Y \subseteq \Bbb A^{2n}$ is an affine variety, given that $X,Y \subseteq \Bbb A^n$ are affine varieties. Is this proof correct? Since both $X$ and $Y$ are affine ...
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### Open subset in Irreducible Topological Space is dense.

Show that every non-empty open subset of an irreducible topological space is dense. I know a lemma that states that $U \subset$X is dense iff for all $A \in \tau$, $A \cap U \neq \emptyset$. So ...
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### Closed sets in Zariski Topology

What are the closed sets on $\mathbb{Z}$, $\mathbb{R}[x]$, $\mathbb{Z}[x]$? I was given this question, but it seems trivial, because aren't the closed sets only the affine varieties? So for each is ...
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### Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
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### Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
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### Irreducible space with infinitely many irreducible components

It would be intuitively satisfying to say the following: A topological space is irreducible if and only if it has exactly one irreducible component. But it is not immediately clear how to prove ...
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### Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
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### If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.

If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen. This is exercise 3.6P of Vakil. I can see that a union of connected components is closed. This is ...
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### Noetherian topological subspaces

I'm trying to prove that any subset of a noetherian topological space is noetherian in its induced topology. MY ATTEMPT OF SOLUTION Let $X$ be a topological space and $Y$ a subspace of $X$. If ...
Probabily it's trivial but I've no idea for a proof. Let $f: X \rightarrow Y$ a continuous map between Topological Spaces, with $Im(f)$ closed in $Y$. I know there exist a covering $\{Y_i\}$ of $Y$ ...
Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...