In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
general topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is said to be orthocompact if every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
has an interior-preserving open
refinement.
That is, given an open cover of the topological space, there is a refinement that 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 definitionally open. That is, every point-finite open cover is interior preserving.
Hence, we have the following: every
metacompact space
In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an ...
, and in particular, every
paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
, 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'' ×
,1of the
closed unit interval with an orthocompact space ''X'' is orthocompact if and only if ''X'' is
countably metacompact. (B.M. Scott)
[ B.M. Scott, Towards a product theory for orthocompactness, "Studies in Topology", N.M. Stavrakas and K.R. Allen, eds (1975), 517–537.]
* Every orthocompact space is countably orthocompact.
* Every countably orthocompact
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
is orthocompact.
See also
*
References
* P. Fletcher, W.F. Lindgren, ''Quasi-uniform Spaces'', Marcel Dekker, 1982, . Chap.V.
Compactness (mathematics)
Properties of topological spaces
External links
*
orthocompact space on
nLab
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