Stronger Forms of Connectedness
There are stronger forms of connectedness for topological spaces, for instance:
- If there exist no two disjoint non-empty open sets in a topological space, X, X must be connected, and thus hyperconnected spaces are also connected.
- Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. Note however, that if the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
- Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected.
In general, note that any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above mentioned topologist's sine curve.
Read more about this topic: Connected Space
Famous quotes containing the words stronger and/or forms:
“The humblest citizen of all the land, when clad in the armor of a righteous cause, is stronger than all the hosts of error.”
—William Jennings Bryan (1860–1925)
“You may melt your metals and cast them into the most beautiful moulds you can; they will never excite me like the forms which this molten earth flows out into. And not only it, but the institutions upon it are plastic like clay in the hands of the potter.”
—Henry David Thoreau (1817–1862)