Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

Why is it that if a space is not Hausdorff and you take the intersection of nested compact subspaces the intersection could be empty? Could you give me an example?

share|cite|improve this question

Take $X$ to be any set with the trivial topology (only the empty set and $X$ are open). Every subset of $X$ is compact. Now take any chain of subsets whose intersection is empty.

For example, $\mathbb N$ with the trivial topology and intersect $A_k = \{n\in\mathbb N\mid n>k\}$.

This idea transfer to any space that has the property that every subspace is compact (this is known as a Noetherian space). For example co-finite topologies have this property.

share|cite|improve this answer
Thank you for the help – vic Dec 3 '12 at 2:36
If the space is non-Hausdorff, what can we say about the nested intersection of connected subsets? Is the intersection connected? – user62775 Apr 21 '13 at 18:57
@upaudel: I don't know. You should ask this as a new question perhaps. – Asaf Karagila Apr 21 '13 at 19:15

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.