Supremum and Infimum are Unique

Theorem

Supremum is Unique

Let $\struct {S, \preceq}$ be an ordered set.

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one supremum in $S$.


Infimum is Unique

Let $\struct {S, \preceq}$ be an ordered set.

Let $T$ be a non-empty subset of $S$.

Then $T$ has at most one infimum in $S$.


Sources

  • 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Orderings
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Theorem $14.3$
  • 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
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