- #1

jwqwerty

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If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above.

The 'Least Upper Bound Property' states that:

If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

My question is that why doesn't the definition of 'Bounded Above' include E≠Φ? Is there a problem when E=Φ? If not, then why does it matter when E=Φ for the 'Least Upper Bound Property'?