How is the region between two concentric spheres simply connected?

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Discussion Overview

The discussion revolves around the concept of simply connected regions in the context of multivariable calculus, specifically focusing on the region between two concentric spheres. Participants explore the conditions under which this region is considered simply connected and address potential misunderstandings related to the boundaries of the spheres.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant expresses confusion about how the region between two concentric spheres can be simply connected, particularly when considering the boundary of the interior sphere.
  • Another participant suggests that once a curve reaches the boundary of the interior sphere, it can slide along that surface to shrink to a point, highlighting the additional dimension available for this process.
  • A further participant acknowledges their initial misunderstanding by noting that they were restricting the curve to a specific equator, and they extend the idea to a spiral surface, suggesting that closed curves can be drawn anywhere on the spiral and can also slide to shrink.

Areas of Agreement / Disagreement

The discussion shows a progression from confusion to clarification, with participants generally agreeing on the mechanics of shrinking curves within the region, though initial misunderstandings about restrictions remain evident.

Contextual Notes

Participants do not fully explore the implications of their statements regarding the nature of simply connected regions, and there may be assumptions about the dimensionality and properties of the surfaces involved that are not explicitly stated.

quasar_4
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Hi,

I've been reviewing multivariable calculus, which I took ages ago, and trying to understand the concept of a simply connected region. The book I'm reading discusses how the region between two concentric spheres is simply connected, but I'm having trouble seeing it. If I think about that region, and imagine sticking a curve there and shrinking it down, don't I run into trouble when I hit the boundary of interior sphere? Can anyone explain to me how this region is simply connected?

Thanks.
 
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Once the curve hits the boundary of the interior sphere it can simply slide along that surface to shrink to a point. It doesn't have to just sit at an equator. That extra dimension in which to shrink is what makes it different from the space between two concentric circles in the plane.
 
Ok, I see. I was trying to restrict it to a given equator. The same idea then applies to the spiral surface - you can draw your closed curve anywhere on the spiral and let it slide down the spiral to shrink it, right?
 
Yes.
 

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