Hyperconnected space
Encyclopedia
In mathematics
, a hyperconnected space is a topological space
X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry
.
The (nonempty) open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects. Thus, a hyperconnected space cannot be Hausdorff
unless it contains only a single point.
Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology
on an algebraic variety
.
Every hyperconnected space is both connected
and locally connected (though not necessarily path-connected or locally path-connected). The continuous image of a hyperconnected space is hyperconnected. In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space is pseudocompact
.
Every open subspace of a hyperconnected space is hyperconnected. A closed subspace need not be hyperconnected, however, the closure
of any hyperconnected subspace is always hyperconnected.
in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.
Unlike the connected component
s of a space, the irreducible components need not be disjoint (i.e. they need not form a partition
). In general, the irreducible components will overlap. Since every irreducible space is connected, the irreducible components will always lie in the connected components.
The irreducible components of a Hausdorff space are just the singleton sets.
Every noetherian topological space
can be written as a finite union of irreducible components.
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...
, a hyperconnected space is 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...
X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
The (nonempty) open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects. Thus, a hyperconnected space cannot be Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
unless it contains only a single point.
Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
.
Every hyperconnected space is both connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
and locally connected (though not necessarily path-connected or locally path-connected). The continuous image of a hyperconnected space is hyperconnected. In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space is pseudocompact
Pseudocompact space
In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded.-Properties related to pseudocompactness:...
.
Every open subspace of a hyperconnected space is hyperconnected. A closed subspace need not be hyperconnected, however, the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of any hyperconnected subspace is always hyperconnected.
Irreducible components
An irreducible componentIrreducible component
In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equationis the union of the two linesandThe notion of irreducibility is stronger than connectedness.- Definition :...
in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.
Unlike the connected component
Connected component
Connected components are part of topology and graph theory, two related branches of mathematics.* For the graph-theoretic concept, see connected component .* In topology: connected component .Implementations:...
s of a space, the irreducible components need not be disjoint (i.e. they need not form a partition
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
). In general, the irreducible components will overlap. Since every irreducible space is connected, the irreducible components will always lie in the connected components.
The irreducible components of a Hausdorff space are just the singleton sets.
Every noetherian topological space
Noetherian topological space
In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets...
can be written as a finite union of irreducible components.