Complete lattices on Q and R ordered by ≤ Ask

manpreet Best Answer 2 years ago

To quote from my lecture notes:

When every subset of A has a lub and glb, we say that the order is a complete lattice, but this takes us beyond the syllabus. It is notable that QQ, ordered by ≤≤, is not a complete lattice but RR, ordered by ≤≤, is a complete lattice. This is the fundamental difference between QQ and RR.

Please can someone explain why this is true? I can't see what the lub of RR would be, in the same way I can't see a lub for QQ.

manpreet 2 years ago

The book is mistaken; for instance, A=R⊆RA=R⊆R has no g.l.b. or l.u.b. in RR, so RR is not a complete lattice. However, Rˆ=R∪{−∞,∞}R^=R∪{−∞,∞} is a complete lattice, where we declare −∞<r<∞−∞<r<∞ for all r∈Rr∈R.

But even this modification doesn't help in the case of Qˆ=Q∪{−∞,∞}Q^=Q∪{−∞,∞}.

Consider the set

Does it have a g.l.b. in