(Partially) ordered topology Also called: partially ordered spaces (pospaces).
Usually ordered topology is used for a topology on linear ordered spaces, where the open intervals are open sets. This is a generalization as for each linear order where open interals are open sets, the order relation is closed.
Topologies generated by the open intervals.
This is restricted to linear orders. Only then it is guaranteed that they are also a ordered topology.
Also known as squeeze or sandwich theorem.
A compact set is bounded below
A compact set is bounded above
A continuous monotone function sends supremum to supremum for nonempty sets.
A continuous monotone function sending bot to bot sends supremum to supremum.
A continuous monotone function sends indexed supremum to indexed supremum.
A continuous monotone function sends infimum to infimum for nonempty sets.
A continuous monotone function sending top to top sends infimum to infimum.
A continuous monotone function sends indexed infimum to indexed infimum.
A continuous monotone function sends supremum to supremum in conditionally complete lattices, under a boundedness assumption.
A continuous monotone function sends indexed supremum to indexed supremum in conditionally complete lattices, under a boundedness assumption.
A continuous monotone function sends infimum to infimum in conditionally complete lattices, under a boundedness assumption.
A continuous monotone function sends indexed infimum to indexed infimum in conditionally complete lattices, under a boundedness assumption.
The extreme value theorem: a continuous function realizes its minimum on a compact set
The extreme value theorem: a continuous function realizes its maximum on a compact set
If the liminf and the limsup of a filter coincide, then this filter converges to their common value, at least if the filter is eventually bounded above and below.
If a filter is converging, its limsup coincides with its limit.
If a filter is converging, its liminf coincides with its limit.
If the liminf and the limsup of a function coincide, then the limit of the function exists and has the same value
If a function has a limit, then its limsup coincides with its limit
If a function has a limit, then its liminf coincides with its limit