I'm reading some notes in which following remark is given:
The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not equal to $ \mathbb A^n$ satisfies at least one non trivial polynomial equation and has therefore necessarily dimension less than $n$, so the closed subsets in Zariski topology are in a sense "very small".
My questions are the following:
- What is the meaning of dimension here?
- What is the meaning of 'so the closed subsets in Zariski topology are in a sense "very small"'?
- What are some other "weird" properties of the Zariski topology?