Monotonically metacompact spaces were recently introduced as an extension of the concept of monotonically compact spaces. In this note we answer a question of Popvassilev, and Bennett, Hart, and Lutzer, by showing that every compact, Hausdorff, monotonically (countably) metacompact space is metrizable. We also show that certain countable spaces fail to be monotonically (countably) metacompact.

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skĭı and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in [ω] , and a version of an (ω, 1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a pos...

All metaLindelöf, and most countably paracompact, homogeneous manifolds are Hausdorff. Metacompact manifolds are never rigid. Every countable group can be realized as the group of autohomeomorphisms of a Lindelöf manifold. There is a rigid foliation of the plane.

In this work, we intend to study metacompact spaces and locally spaces,and their properties relations with order topological spaces.Several examples are discussed many wellknown theorems generalized concerning [7, 2, 8, 3, 4, 56].