Compact directed spaces

Ward, L. E.

doi:10.1090/S0002-9947-1970-0268858-2

Compact directed spaces
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Trans. Amer. Math. Soc. 152 (1970), 145-157 Request permission

Abstract:

A directed space is a partially ordered topological space in which each two elements have a common predecessor. It is a consequence of a theorem of A. D. Wallace that a compact directed space is acyclic if each of its principal ideals is acyclic. This result is extended by considering the situation where at most finitely many principal ideals are not acyclic. It turns out that some of the elements which generate nonacyclic principal ideals must be maximal and that the $p$th cohomology group of the space must contain the $p$th cohomology group of such a principal ideal as a direct summand. In the concluding sections it is shown that these spaces can be made acyclic by dividing out a closed ideal which contains all of the nonacyclic principal ideals, and some results on the acyclicity properties of minimal partial orders on compact spaces are proved.

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