Connected Space - Stronger Forms of Connectedness

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:

    And we can get back to that raw state
    Of feeling, so long deemed
    Inconsequential and therefore appropriate to our later musings
    About religion, about migrations. What is restored
    Becomes stronger than the loss as it is remembered;
    Is a new, separate life of its own.
    John Ashbery (b. 1927)

    Media mystifications should not obfuscate a simple, perceivable fact; Black teenage girls do not create poverty by having babies. Quite the contrary, they have babies at such a young age precisely because they are poor—because they do not have the opportunity to acquire an education, because meaningful, well-paying jobs and creative forms of recreation are not accessible to them ... because safe, effective forms of contraception are not available to them.
    Angela Davis (b. 1944)