Upper topology explained

\{a\}

is the order section

a]=\{x\leqa\}

for each

a\inX.

If

\leq

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.

(-infty,+infty]=\R\cup\{+infty\}

by the system

\{(a,+infty]:a\in\R\cup\{\pminfty\}\}

of open sets. Similarly, the real lower topology

\{[-infty,a):a\in\R\cup\{\pminfty\}\}

is naturally defined on the lower real line

[-infty,+infty)=\R\cup\{-infty\}.

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

{[-infty,+infty)}.

Similarly, a function into the upper real line is lower semi-continuous if and only if it is upper-continuous, i.e. is continuous with respect to the upper topology on

{(-infty,+infty]}.

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Upper topology".

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