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?
Infinite dimensional vector spaces and inductive sets
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abstract-algebra
vector-spaces
axiom-of-choice
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1When you say inductive you mean...? – 2012-11-05
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0I mean that every totally ordered subset of $S$ has an upper bound. – 2012-11-05
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0I never knew that was the meaning of inductive. But very well. – 2012-11-05
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0I 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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0I 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