In topology, a space is connected if it cannot be separated, that is there do not exist disjoint non-empty open sets such that (this is often expressed as ). For example, the set is not connected as a subspace of .
There are several definitions that are related to connectedness:
- is path-connected if for any two points , there exists a continuous function such that . Path-connectedness is a stronger condition that connectedness.
- is locally connected at a point if for any open set about , there exists a connected open set such that . is locally connected if it is locally connected at every single point in . Perhaps surprisingly, local connectedness and connectedness are independent properties.
- Similarly, is locally path-connected at a point if for any open set about , there exists a path-connected open set such that . is locally path-connected if it is locally path-connected at every single point in .
One can build connected spaces using the following properties.
- The image of a connected space under a continuous function is also connected.
- The union of connected spaces that share a point in common is also connected.
- The union of a connected space with its limit points is also connected.
- The finite Cartesian product of a connected space is also connected (under the product topology).
Connectedness can be used to define an equivalence relation on an arbitrary space . If we define equivalence relation if there exists a connected subspace of containing , then the resulting equivalence classes are called the components of . In a sense, the components are the maximally connected subsets of .
We can define path-components in the same manner.
The quasicomponents are the equivalence classes resulting from the equivalence relation if there does not exist a separation such that . This definition is weaker than that of a component, for any component must lie in a quasicomponent (the definitions are equivalent if is locally connected).
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