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?

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    $\begingroup$ 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
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    $\begingroup$ 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.


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