Insertion of poset-valued maps with the way-below and -above relations

In this paper, with the way-below relation ≪ and the way-above relation ≪d, we give a poset-valued insertion theorem by a pair of semi-continuous maps. Moreover, we show a poset-valued insertion theorem by a continuous map as follows: Let X be a paracompact Hausdorff space, P a bi-bounded complete, bicontinuous, pathwise connected, topological poset. For each upper semi-continuous map f:X→P with a lower bound and each lower semi-continuous map g:X→P, if 〈f,g〉 has interpolated points pointwise, there exists a continuous map h:X→P such that f≪dh≪g. A generalized Dowker-Katětov's insertion theorem, by using the way-below and -above relations on bicontinuous posets P, is also given.