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:
“When the masculine mystique is pulling boys and men out into the world to growl manly noises at one another, the only power with a stronger pull on the male psyche is maternally induced guilt. The guilt is quite necessary for our moral development, but it is often uncomfortable.”
—Frank Pittman (20th century)
“No rent-roll nor army-list can dignify skulking and dissimulation: and the first point of courtesy must always be truth, as really all the forms of good-breeding point that way.”
—Ralph Waldo Emerson (1803–1882)