Take the 2-minute tour ×
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.

I'm working through an exercise in Hartschorne concerning direct limits of sheaves on noetherian topological spaces. It seems natural to use the the fact that every open subset of a noetherian topological space is the quasi-compact in its induced topology. I'm some trouble proving what I need when I only use this property. I'm guessing that I'll have to use noetherian-ness more directly, but I thought I would ask if anyone knew an example offhand.

Is there a non-notherian topological space which has this property?

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

If every open subspace of a space is quasi compact, then the space is Noetherian. This is exercise 6.5 in Atiyah Macdonald. It's an easy proof. Let me know if you need more help with it. Hint: If you consider an increasing sequence of open sets, its union is open and hence quasi compact.

share|improve this answer
    
Hey thanks! I knew I should just have thought about this more. I've been rather distracted as of late and haven't really take the time I should to sit down and think about things. –  Jamie Weigandt Nov 16 '10 at 21:47
add comment

No. Noetherianness is equivalent to the condition that every open subset is compact, and is in fact equivalent to the condition that every subset is compact. This is a nice exercise.

share|improve this answer
add comment

Your Answer

 
discard

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.