If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Multivariable calculus>Unit 5

Lesson 10: Types of regions in three dimensions

# Type II regions in three dimensions

Definition and intuition for Type 2 Regions. Created by Sal Khan.

## Want to join the conversation?

• I'm wondering, is every simply connected region in R^3 a Type 1, Type 2, and a Type 3 shape? If so, can we extend this to n-dimensions and say that for regions in R^n, a simply connected region is a Type 1, 2, ..., n region?
• No.

First, I assume that by "simply connected" you are using the usual topological definition that any closed loop can be contracted continuously through the region to a point. So all the examples that Sal uses are simply connected. But, say, a solid torus is not simply connected - consider a loop going all the way round the ring - you can't pull it tight and contract it to a point without crossing the hole in the doughnut, which is not part of the region.

If that isn't what you mean by "simply connected", then you'll have to explain what you do mean as I'm not aware of any other standard usage of that term.

Alternatively, if you are talking about the surface of the region being simply connected, I don't think that changes the situation. The surfaces of all Sal's examples are simply connected. The surface of the torus is not simply connected.

And my gut feeling is that for any bounded, closed surface in R^3 the region it encloses being simply connected is equivalent to the surface being simply connected (unless the region / surface is in some way pathological - so surfaces that aren't piecewise smooth with a finite number of smooth components, fractal surfaces, Klein bottles and the like might not work) - I'd be happy to be contradicted on this, but can't at the moment see a counterexample!

For an example of a region that is not a type 1, a type 2 or a type 3 region, consider a region formed by 3 dumbbells oriented along the x, y and z axes respectively and crossing at the origin - with large enough. bulges / long enough and narrow enough bars connecting them so that the only places where any two of the dumbbells intersect is in around the origin where the connecting bars cross. Note that this IS a simply connected region!