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Concept
Connected space
Summary
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 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 is a if it is a connected space when viewed as a subspace of .
Some related but stronger conditions are path connected, simply connected, and -connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
A topological space is said to be if it is the union of two disjoint non-empty open sets. Otherwise, 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 the following conditions are equivalent:
is connected, that is, it cannot be divided into two disjoint non-empty open sets.
The only subsets of which are both open and closed (clopen sets) are and the empty set.
The only subsets of with empty boundary are and the empty set.
cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other's closure).
All continuous functions from 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 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.
Given some point in a topological space the union of any collection of connected subsets such that each contains will once again be a connected subset.
The connected component of a point in is the union of all connected subsets of that contain it is the unique largest (with respect to ) connected subset of that contains
The maximal connected subsets (ordered by inclusion ) of a non-empty topological space are called the connected components of the space.
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