Intereting Posts

Tall fraction puzzle
Weak categoricity in first order logic
nilpotent group is almost abelian – counterexample for infinite order case
Prime Arithmetic Progression with one fixed Element
Expected Ratio of Coin Flips
Limit value of a product martingale
Non-Symmetric Positive Definite Matrices
parity bias for trees?
Show that there are infinitely many powers of two starting with the digit 7
How to construct symmetric and positive definite $A,B,C$ such that $A+B+C=I$?
How common is it for a densely-defined linear functional to be closed?
Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?
Intuitive explanation for $\mathbb{E}X= \int_0^\infty 1-F(x) \, dx$
Show $\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$ for $t \gt 0$
An equivalent definition of the profinite group

Question:Let $X$ be an affine variety over $\Bbb C$, and let $Y\subseteq X$ be a constructible set (i.e. $Y$ is a finite union of locally closed sets). Is it true that the Zariski closure of $Y$ is the same as the closure of $Y$ in the standard Euclidean topology inherited from the inclusion $X\subseteq\Bbb C^n$?

In this question I asked whether these two closures were the same when $Y$ is the orbit of an algebraic group action. Then, this answer says that the answer is yes because orbits are constructible sets. However, I don’t know a proof of the fact that these orbit closures are the same for constructible sets.

If $\bar{Y}^E$ denotes the euclidean closure and $\bar{Y}^Z$ the Zariski one, then it is clear that

$$\bar{Y}^E\subseteq \bar{Y}^Z$$

since the Euclidean topology is finer than the Zariski topology. But for the converse, I am clueless.

- Embedding of Kähler manifolds into $\Bbb C^n$
- Can manifolds be uniformly approximated by varieties?
- Graph of morphism , reduced scheme.
- The ring of germs of functions $C^\infty (M)$
- Division algorithm of multivariate polynomial
- closed immersion onto an affine scheme - showing affineness

- Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$
- Continuous $f\colon \to \mathbb{R}$ all of whose nonempty fibers are countably infinite?
- Limit points in topological space $X$
- Configuration scheme of $n$ points
- Correspondences between Borel algebras and topological spaces
- Two-sheeted covering of the Klein bottle by the torus
- Your favourite application of the Baire Category Theorem
- Connected, locally connected, path-connected but not locally path-connected subspace of the plane
- Longest sequence of minimally finer topologies
- Polynomial map is surjective if it is injective

Yes it is true. You probably know that a non-empty Zariski-open set $U\subseteq X$ is Zariski-dense, and it is a standard fact that $U$ is also Euclidean-dense. (See this mathoverflow answer for a slick proof of that fact.) Assuming this, the proof of your claim is simple:

Wlog, $Y$ is locally closed because the closure of a finite union is the union of the closures. Then, by definition, $Y$ is Zariski-open in $\bar{Y}^Z$ and hence $Y$ is Euclidean-dense in $\bar{Y}^Z$, i.e. $\bar{Y}^E\cap\bar{Y}^Z=\bar{Y}^Z$. Since $\bar{Y}^E\subseteq\bar{Y}^Z$, this concludes the proof.

- Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$
- $A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated?
- There are apparently $3072$ ways to draw this flower. But why?
- Example for fintely additive but not countably additive probability measure
- Prove $' = \cos x$ without using $\lim\limits_{x\to 0}\frac{\sin x}{x} = 1$
- Prove that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer without using concepts of abstract algebra
- Techniques to prove a function is uniformly continuous
- Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic
- Does weak compactness imply boundedness in a normed vector space (not necessarily complete)?
- Cramer's rule: Geometric Interpretation
- How many solutions has the equation $\sin x= \frac{x}{100}$ ?
- The automorphism group of the real line with standard topology
- How to find the original coordinates of a point inside an irregular rectangle?
- Relation between $\gcd$ and Euler's totient function .
- Fascinating Lampshade Geometry