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
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