Orthocompact space
Encyclopedia
In 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. That is, given an open cover of the topological space, there is a refinement which is also an open cover, with the further property that at any point, the intersection of all open sets in the refinement containing that point, is also open.
If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every point finite open cover is interior preserving. Hence, we have the following: every metacompact space
, and in particular, every paracompact space
, is orthocompact.
Useful theorems:
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 orthocompact if every open cover has an interior preserving open refinement. That is, given an open cover of the topological space, there is a refinement which is also an open cover, with the further property that at any point, the intersection of all open sets in the refinement containing that point, is also open.
If the number of open sets containing the point is finite, then their intersection is clearly open. That is, every point finite open cover is interior preserving. Hence, we have the following: every metacompact space
Metacompact space
In 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 in particular, every paracompact space
Paracompact space
In 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 orthocompact.
Useful theorems:
- Orthocompactness is a topological invariant; that is, it is preserved by homeomorphisms.
- Every closed subspace of an orthocompact space is orthocompact.
- A topological space X is orthocompact if and only if every open cover of X by basic open subsets of X has an interior-preserving refinement that is an open cover of X.
- The product X × [0,1] of the closed unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
with an orthocompact space X is orthocompact if and only if X is countably metacompact. (B.M. Scott) - Every orthocompact space is countably orthocompact.
- Every countably orthocompact Lindelöf space is orthocompact.