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Consider an infinite dimensional vector space $E$ and define $$S:=\left\{F \subset E\biggr| F\ne 0\text{ is a subspace of }E\right\}.$$ Endow $S$ with the reverse inclusion. Is it possible to find a space $E$ for which $S$ is not inductive?

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    When you say inductive you mean...?2012-11-05
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    I mean that every totally ordered subset of $S$ has an upper bound.2012-11-05
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    I never knew that was the meaning of inductive. But very well.2012-11-05
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    I adopt the definition given in Brezis' "Functional Analysis, Sobolev spaces and Partial Differential Equation", but if you say so I'm pretty sure that is not an universal convention!2012-11-05
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    I see. I can understand the origin of this terminology, but for future reference it is best to give a definition (in particular when the definition is very short, and you are pretty sure there is no universal convention). I hope my answer is to your liking.2012-11-05

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