In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton is the order section for each If is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets. However, not all up-sets must necessarily be open sets. The lower topology induced by the preorder is defined similarly in terms of the down-sets. The preorder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder.
is the order section ![{\displaystyle a]=\{x\leq a\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4e5b78692ec4903b98eceac64a5132ae1982e9)
for each 
If 
is a partial order, the upper topology is the least The real upper topology is most naturally defined on the upper-extended real line by the system of open sets. Similarly, the real lower topology is naturally defined on the lower real line A real function on a topological space is upper semi-continuous if and only if it is lower-continuous, i.e. is continuous with respect to the lower topology on the lower-extended line
![{\displaystyle (-\infty ,+\infty ]=\mathbb {R} \cup \{+\infty \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/741d2c6b6c93a8bd6a918a624fa637837735a892)
by the system ![{\displaystyle \{(a,+\infty ]:a\in \mathbb {R} \cup \{\pm \infty \}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7793f14b3d97d6cd2a914c8ed53444cde0eeab7)
of open sets. Similarly, the real lower topology 
is naturally defined on the lower real line 
A real function on a 
![{\displaystyle {(-\infty ,+\infty ]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a899d77101aa283791fddd0391545542772470)
See also
- List of topologies – List of concrete topologies and topological spaces
References
- Gerhard Gierz; K.H. Hofmann; K. Keimel; J. D. Lawson; M. Mislove; D. S. Scott (2003). Continuous Lattices and Domains. Cambridge University Press. p. 510. ISBN 0-521-80338-1.
- Kelley, John L. (1975) . General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153. p.101
- Knapp, Anthony W. (2005). Basic Real Analysis. Birkhhauser. p. 481. ISBN 0-8176-3250-6.
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