A topological space $X$ is called quasi-compact if whenever ${(U_i)}_{i∈S}$ are a family of open subsets such that $∪_{i∈S}(U_i) = X$ then there are a finite number of $(U_i)$’s which actually cover X. I’m thinking first we show that if $(U_{fi})$, $i ∈ S$ (where ${f_i}$, $i ∈ S$ a collection of elements of $R[X]$) […]

How to show that closure of image by polynomial of irreducible algebraic variety is also irreducible algebraic variety. $A$ -irreducible algebraic variety, $F$ – polynomial map ($F:A \to B$), proof that $\bar B$ – is irreducible algebraic variety. Closure of some set is always algebraic variety as definition of closure. But how to show, that […]

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that $M=P\,\text{diag}(1,\dots,1,-1)P^{-1}$ is involutory ($MM=I$) and $\det(M)=-1$. Possible starting point: Consider $$\{P\in GL_n(\mathbb R),\ D \text{ invertible and diagonal s.t. } DP\text{diag}(1,\dots,1,-1)P^{-1}D^{-1} \text{ is orthogonal}\}$$ In general, this […]

In topology and analysis we define the support of a continuous real function $f:X\rightarrow \mathbb R$ to be $ \left\{ x\in X:f(x)\neq 0\right\}$. This is the complement of the fiber $f^{-1} \left\{0 \right\}$. So it looks like the support is always an open set. Why then do we take its closure? In algebraic geometry, if […]

Let $(\mathfrak a_i)$ be an infinite family of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ not defined? I am trying to understand Zariski topology. Here, $V(\bigcap_i \mathfrak a_i)= \bigcup\limits_{i} V(\mathfrak a_i)$. If $\bigcap\limits_{i=1}^\infty \mathfrak a_i$ is defined, which I think it should be, then we would have an infinite union of sets of […]

I would like to ask whether given a topological space $X$, we can find a commutative ring with unity $R$ such that $\operatorname{Spec} R$ (together with the Zariski topology) is homeomorphic to $X$. Since the spectrum is a compact space, this is obviously only possible if $X$ is compact. Furthermore, from this answer we obtain […]

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn’t satisfy the Hausdorff separation axiom. Ok the basis is very simple, but what are the advantages?

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