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I want to prove that an irreducible affine variety is connected in the Zariski topology. Actually this is an exercise from Dummit and Foote's Abstract Algebra. However, I have two questions:

  • Is the nonempty open set mentioned in problem 12(also mentioned in problem 11) corresponding to the whole space $\Bbb A^n$ or the variety itself? It's hard for me to figure out what author mean.
  • I have tried to think about it for a long while. My attempt was proof by contradiction(I assumed the nonempty "open" sets is open with respect to the whole space $\Bbb A^n$). Let $V$ be an irreducible affine variety, $A,~B$ be open in $\Bbb A^n$ and $V\subseteq A\cup B$, $A\cap B=\emptyset$, $A\cap V\neq\emptyset$, and $B\cap V\neq\emptyset$. I want to deduce a contradiction, but fail.

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    $\begingroup$ The Zariski topology on an affine variety is the same as the subspace topology for the variety as a subset of $\mathbb{A}^n$ with the Zariski topology. I think this is why they are a little loose with saying where the opens are open. $\endgroup$
    – user113102
    Commented May 14, 2019 at 15:50
  • $\begingroup$ As user113102 pointed out, it doesn't really matter where the open sets live. However, to clarify the problem slightly, the fact you are given is that $A\cap V$ is dense in $V$, see problem 11. If you use problem 11, you are one sentence away from a contradiction. $\endgroup$
    – jgon
    Commented May 14, 2019 at 18:53

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Your question is purely topological and has nothing to do with algebraic varieties.
$\bullet$ A topological space is said to be connected if it is impossible to write it as the union of two disjoint non-empty open subsets.
$\bullet \bullet$ A topological space is said to be irreducible if it is non-empty and if any two non-empty open subsets have non-empty intersection connected.

It is then obvious that every irreducible topological space is connected.
The simplest connected but not irreducible topological space is $X=\{f,o,o'\}$ endowed with the topology whose open sets are: $$X,\emptyset, \{o\},\{o'\},\{o,o'\}.$$
The celebrated Professor Grossmaulkleinesgehirn has proved the fundamental classification theorem:
The irreducible Hausdorff topological spaces are exactly the one point spaces.

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