Supremum and Infimum are Unique
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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): $\S 7$