Upper topology

 ( General Topology ) 

In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton $\backslash \{a\backslash \}$ is the order section $a]\; =\; \backslash \{x\; \backslash leq\; a\backslash \}$ for each $a\backslash in\; X.$ If $\backslash leq$ is a partial order, the upper topology is the least order consistent topology in which all are . However, not all up-sets must necessarily be open sets. The lower topology induced by the preorder is defined similarly in terms of the . The preorder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder.

The real upper topology is most naturally defined on the upper-extended real line $(-\backslash infty,\; +\backslash infty]\; =\; \backslash R\; \backslash cup\; \backslash \{+\backslash infty\backslash \}$ by the system $\backslash \{(a,\; +\backslash infty]\; :\; a\; \backslash in\; \backslash R\; \backslash cup\; \backslash \{\backslash pm\backslash infty\backslash \}\backslash \}$ of open sets. Similarly, the real lower topology $\backslash \{-\backslash infty,\}.$

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See also

• Gerhard Gierz (2025). 9780521803380, Cambridge University Press. . ISBN 9780521803380
• p. 101
• Anthony W. Knapp (2025). 9780817632502, Birkhhauser. ISBN 9780817632502

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