Find a set which has GLB and LUB but there exists at least one subset which has no GLB and LUB

GLB=greatest lower bound LUB=least upper bound Give one example of a set such that the GLB and LUB exist but there exists at least one subset which has no GLB and LUB.

• Hmm, is the set supposed to be a set of reals? The real-analysis tag suggests that it is, but boolean-algebra could imply that you're thinking about a more general order-theoretic setting. – Henning Makholm Jan 23 '17 at 12:44

Consider the set $\{42\}$. It has a greatest lower bound and a least upper bound (both of which are $42$), but its subset $\varnothing$ has neither.
• @астонвіллаолофмэллбэрг: Obviously not -- but for example in $\mathbb Q$ we could consider $\{x\mid -2<x<2\}$ and its subset $\{x\mid x^2<2\}$. (That subset has a GLB and LUB in $\mathbb R$, of course, but not if $\mathbb Q$ is all we know). – Henning Makholm Jan 23 '17 at 12:51
Consider the partially ordered set $$P=\{1,2,3,5,12,18,72,108,1080\}$$ ordered by divisibility. The set $X=\{5,12,18\}$ has greatest lower bound $1$ and least upper bound $1080,$ but its subset $Y=\{12,18\}$ has neither a greatest lower bound nor a least upper bound in $P.$