Formal Definition
A topological space X is said to be disconnected if it is the union of two disjoint nonempty 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.
- X cannot be divided into two disjoint nonempty closed sets.
- The only subsets of X which are both open and closed (clopen sets) 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 nonempty separated sets.
- The only continuous functions from X to {0,1} are constant.
Read more about this topic: Connected Space
Famous quotes containing the words formal and/or definition:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)