Mesocompact space
Encyclopedia
In mathematics
, in the field of general topology
, a topological space
is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many members of the refinement.
The following facts are true about mesocompactness:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any open cover, we can find an open refinement with the property that every compact set meets only finitely many members of the refinement.
The following facts are true about mesocompactness:
- Every compact spaceCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, and more generally every paracompact spaceParacompact spaceIn mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
is mesocompact. This follows from the fact that any locally finiteLocally finite collectionIn the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....
cover is automatically compact-finite. - Every mesocompact space is metacompactMetacompact spaceIn mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
, and hence also orthocompactOrthocompact spaceIn mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior preserving open refinement...
. This follows from the fact that points are compact, and hence any compact-finite cover is automatically point finite.