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:
“Our rural village life was a purifying, uplifting influence that fortified us against the later impacts of urbanization; Church and State, because they were separated and friendly, had spiritual and ethical standards that were mutually enriching; freedom and discipline, individualism and collectivity, nature and nurture in their interaction promised an ever stronger democracy. I have no illusions that those simpler, happier days can be resurrected.”
—Agnes E. Meyer (1887–1970)
“Your letter is come; it came indeed twelve lines ago, but I
could not stop to acknowledge it before, & I am glad it did not
arrive till I had completed my first sentence, because the
sentence had been made since yesterday, & I think forms a very
good beginning.”
—Jane Austen (1775–1817)