In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
general topology
In mathematics, general 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 differential topology, geomet ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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 collection 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\s ...
has an interior-preserving open
refinement
Refinement may refer to: Mathematics
* Equilibrium refinement, the identification of actualized equilibria in game theory
* Refinement of an equivalence relation, in mathematics
** Refinement (topology), the refinement of an open cover in mathem ...
.
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 clearly 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 open ...
, 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
.
* 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
In 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. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
with an orthocompact space ''X'' is orthocompact if and only if ''X'' is
countably metacompact
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 open ...
. (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
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