# Are there non-noetherian topological spaces in which every open subset is quasi-compact?

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?

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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.

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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

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.

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