and related branches of mathematics
, a connected space is a topological space
that cannot be represented as the union
of two or more disjoint non-empty open subsets
. Connectedness is one of the principal topological properties
that are used to distinguish topological spaces.
A subset of a topological space ''X'' is a connected set if it is a connected space when viewed as a subspace
Some related but stronger conditions are path connected
, simply connected
, and n-connected
. Another related notion is locally connected
, which neither implies nor follows from connectedness.
A topological space
''X'' is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, ''X'' is said to be connected. A subset
of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set
(with its unique topology) as a connected space, but this article does not follow that practice.
For a topological space ''X'' the following conditions are equivalent:
#''X'' is connected, that is, it cannot be divided into two disjoint non-empty open sets.
#''X'' cannot be divided into two disjoint non-empty closed set
#The only subsets of ''X'' which are both open and closed (clopen set
s) are ''X'' and the empty set.
#The only subsets of ''X'' with empty boundary
are ''X'' and the empty set.
#''X'' cannot be written as the union of two non-empty separated sets
(sets for which each is disjoint from the other's closure).
functions from ''X'' to
are constant, where
is the two-point space endowed with the discrete topology.
Historically this modern formulation of the notion of connectedness (in terms of no partition of ''X'' into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz
, and Felix Hausdorff
at the beginning of the 20th century. See for details.
connected subsets (ordered by inclusion
) of a non-empty topological space are called the connected components of the space.
The components of any topological space ''X'' form a partition
of ''X'': they are disjoint
, non-empty, and their union is the whole space.
Every component is a closed subset
of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational number
s are the one-point sets (singletons
), which are not open.
be the connected component of ''x'' in a topological space ''X'', and
be the intersection of all clopen
sets containing ''x'' (called quasi-component
of ''x''.) Then
where the equality holds if ''X'' is compact Hausdorff or locally connected.
A space in which all components are one-point sets is called totally disconnected
. Related to this property, a space ''X'' is called totally separated if, for any two distinct elements ''x'' and ''y'' of ''X'', there exist disjoint open sets
''U'' containing ''x'' and ''V'' containing ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff
, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
* The closed interval