Pointfree functional compactness (Q2460681)

Under compactification of a space \(X\) we understand an embedding of \(X\) onto a dense subset of a compact Hausdorff space. For any form of weak compactness one can also ask for the existence of an embedding onto a dense subset of the ambient space which has the considered form of weak compactness. Functional compactness is treated in the paper. The author has found two classes of spaces that admit embeddings onto a dense subset of a functionally compact space: If \(X\) is either discrete, or \(\theta\)-seminormal and has finitely many free open ultrafilters, each one being regular, then \(X\) can be densely embedded in a functionally compact space. Proofs are via pointfree topology. First, functional compactness is translated into frame-theoretic language (pointfree functional compactness). Then an adequate frame techique is developed. One of the proved statements says that a frame \(L\) is functionally compact iff \(L\) is almost compact and \(\theta\)-seminormal. Since the Katětov extension of any frame is almost compact, the class of frames is characterized that have \(\theta\)-seminormal Katětov extension. All this together, translated into space topology, gives the quoted results. In addition, the concept of rigidity of subspaces of topological spaces is introduced in frame-theoretic language and several results are proved.