Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with handle) is simply connected, but a hollow rubber ball is simply connected.

In two dimensions, a circle is not simply connected, but a disk and a line are (not sure how a line is, but ok). Spaces that are connected but not simply connected are called non-simply connected or multiply connected.

Is the sphere the only 3d object that can be a truly connected space?

Or is this more aptly used in 1D spaces?

  • 1
    $\begingroup$ What do you mean by "truly connected"? Do you mean "connected and simply connected"? $\endgroup$ – Arthur Mar 14 at 21:45
  • $\begingroup$ yes - sorry simply connected $\endgroup$ – Rick Mar 14 at 21:46
  • 2
    $\begingroup$ Well, a disc can lie in three-dimensional space, and it is just as simply connected there. $\endgroup$ – Arthur Mar 14 at 21:51
  • $\begingroup$ but only on 1 dimension. If the connection goes around the flat side of the disc, it is not simply connected anymore. This would be like a torus, wouldn't it? $\endgroup$ – Rick Mar 14 at 22:17
  • $\begingroup$ $R^n$ minus a finite amount of points is simply connected for $n \geq 3$. So you could take any open ball in $R^3$ minus a finite amount of points, and that will be simply connected. They are not homeomorphic to a sphere. $\endgroup$ – Pel de Pinda Mar 15 at 15:07

$\mathbb{R}^3$ is simply connected, and not homeomorphic to the sphere.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.