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:
“We cannot be any stronger in our foreign policy—for all the bombs and guns we may heap up in our arsenals—than we are in the spirit which rules inside the country. Foreign policy, like a river, cannot rise above its source.”
—Adlai Stevenson (1900–1965)
“It is given to few to add the store of knowledge, to strike new springs of thought, or to shape new forms of beauty. But so sure as it is that men live not by bread, but by ideas, so sure is it that the future of the world lies in the hands of those who are able to carry the interpretation of nature a step further than their predecessors.”
—Thomas Henry Huxley (1825–95)