Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider the projective space ${\mathbb P}^{n}_{k}$ with field $k$. We can naturally give this the Zariski topology.

Question: What are the (proper) compact sets in this space?

Motivation: I wanted nice examples of spaces and their corresponding compact sets; usually my spaces are Hausdorff and my go-to topology for non-Hausdorff-ness is the Zariski topology. I wasn't really able to find any proper compact sets which makes me think I'm doing something wrong here.

share|cite|improve this question
$\mathbb{P}^n_k$ is a Noetherian topological space, so every subset is (quasi)compact. Moreover every closed subset is complete (the analogue of compactness in algebraic geometry). – Zhen Lin Nov 18 '11 at 17:04
Everything Zhen writes in his great comment is absolutely correct and relevant:+1. Readers not used to algebraic geometry (this excludes Zhen and many others!) should beware, however, that completeness is not a purely topological notion, but depends on the scheme-theoretic structure. For example $\mathbb A^1_k$ is not complete but is homeomorphic to $\mathbb P^1_k$ which is complete. – Georges Elencwajg Nov 18 '11 at 17:37
up vote 5 down vote accepted

You are in for a big surprise, james: every subset of $\mathbb P^n_k$ is quasi-compact.
This is true more generally for any noetherian space, a space in which every decreasing sequence of closed sets is stationary.
However: the compact subsets of $\mathbb P^n_k$ are the finite sets of points such that no point is in the closure of another.

A topological space $X$ is quasi-compact if from every open cover of $X$ a finite cover can be extracted. A compact space is a Hausdorff quasi-compact space.

Bibliography Bourbaki, Commutative Algebra, Chapter II, §4,2.

share|cite|improve this answer
Ah, this is nice. I kept seeing "quasi-compact" but I mistook the definition! You note then that there are compact subsets though --- what is the difference between the compact subsets and the quasi-compact subsets? – james Nov 19 '11 at 1:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.