CHRISTINE BREINER: Welcome back to recitation.
In this video, what I'd like us to do is work on
understanding simply connected regions in three dimensions.
Well, there's one two-dimensional one, but the
rest are three dimensions.
So what I want you to do is for each of the following--
there are six different regions--
determine whether or not each of them is simply connected.
So the first one is R3.
The second one is if I take R3 and I
remove the entire z-axis.
The third one is if I take R3 and I remove 0.
The fourth one is if I take R3 and remove a circle.
The fifth one is R2 minus a line segment.
And the sixth one is a solid torus.
So a solid torus looks like a doughnut, and it includes the
inside of the doughnut.
This looks like a doughnut, hopefully, to you.
And it's not hollow.
It includes the inside.
So what I'd like you to do, again, is determine whether or
not each of these regions is simply connected.
And why don't you pause the video while you work on that.
And then bring the video back up when you're ready
to check your work.
OK, welcome back.
So again, what we're interested in doing is
understanding simply
connectedness in another dimension.
We did something already a while back with two
dimensions, and so now we want to understand it better in
three dimensions.
So let's work through these.
Well, I'm not going to write anything down for number one,
because you should already know that
R3 is simply connected.
But if you weren't sure about it, you could think, for any
closed curve I draw in R3, I can certainly get all of the
inside of it contained in R3.
Another way to think about it is that I can take that curve
and I can collapse it down to a point, and remain in R3.
So then the first one is an easy yes to simply
connectedness.
OK?
So let's start on the second one, and I'm going to draw a
little picture for us.
So the second one is R3.
I should go this way.
This is x, y, and z, but then I remove the entire z-axis.
So I should make this really dark so we know we're removing
that part from R3.
And I'm removing it all the way up to minus infinity in
the z-direction and plus infinity in the z-direction.
Now, the question is can I find any closed curve, that
when I try and compress that closed curve down to a point,
I can't do it while remaining inside this region that is all
of R3 minus the z-axis.
And the answer is there is a whole family of
curves that do this.
If I take a curve that goes around the z-axis, you'll
notice that there's something on the inside of it--
regardless, of whether I slide it up or down-- there's a
point on the inside of this curve that is not in the
region I'm interested in.
The region, again, is R3 minus the z-axis.
So there are two ways to think about this.
You can think about if I were to take this curve and I were
to put a surface across this curve so it was like a disk,
there would be a point on the z-axis that
would intersect it.
Or you can think about it as saying, I have this curve and
if I try and squeeze it down to as small as I can get it, I
can't get it as small is I want without hitting the
z-axis at some point.
The z-axis is kind of in the way, right?
Now, number three is a little different situation.
Because in number three, I think this exact same picture,
but instead of removing the whole z-axis, I just remove
the origin.
So let me try and draw a picture of that.
So I'm going to make this a big open circle at the origin.
That's not included in our region.
So our region is all of R3 except the origin.
And in two-dimensional space, this was not simply connected.
But in three-dimensional space it is simply connected.
So this is a little different situation than what you had
previously.
And so the idea is here, if I take a curve, even if I take a
curve that's sitting in the xy plane that goes around the
origin, the point is I can keep this curve in
three-dimensional space, and I can wiggle it around, so that
I can shrink it down to a point, and the origin doesn't
get in the way.
It doesn't keep me from doing that.
So actually, this region, even though in two-dimensional
space it was not simply connected, in
three-dimensional space it is.
And let's see if we understand the difference.
The difference is in two-dimensional space, if I
drew a curve on the xy plane around the origin, and I
wanted to squish it down to a point, the only way to do that
would be to bring the curve somehow through the origin.
Right?
I would be stuck having to pass the curve through the
origin to shrink it down to a point.
But in three space, I have another dimension.
So a curve that sits on the xy plane, I can just kind of lift
it a little bit away from the origin, and then I can shrink
it down to a point without the origin getting in the way.
So having that extra dimension means even though I remove one
point, it's still actually a simply connected region.
So maybe this is the first place we see that in the three
dimensions we have a different case than we had in two
dimensions removing the same kind of object.
So I realize now I haven't been writing down whether
these are simply connected or not.
So I should write down this is simply connected.
And maybe for number two I should go back and formally
write not simply connected.
So that we have this for posterity.
Now the fourth one is R3 minus the circle.
So let me see if I can draw a picture of that.
And for the circle, it doesn't really matter where it is.
I'm just going to draw one somewhere.
So here's my circle.
So everything is in my region except this circle.
And the question: is it simply connected?
And the answer is: no, the region is not simply
connected, because of one particular problem.
It's actually the same kind of problem you have when you
remove the z-axis.
And that is, if I draw a curve that goes around this circle--
any curve that goes around this circle--
notice that any way I try and move this curve and shrink it
down to a point, this circle is going to get in the way for
the same reason that the z-axis got in the way.
Because this circle is closed, I can't slide the curve I'm
interested in away from the circle and
then shrink it down.
OK.
There's some sort of obstruction right here.
And so it's fundamentally different than the case where
we just had the origin, because we could take any
curve and we could move it away from the origin, and then
shrink it down to a point.
And the origin didn't get in the way.
But here, anywhere I try and move this curve, it's going to
have to hit the circle if I want to move it away so I can
shrink it to a point in my region.
So this circle is preventing me from shrinking it down.
OK, and then there are two more.
And the fifth one is R2 minus a line segment.
So now we're in two-dimensional space.
OK, and let me just pick a segment.
OK.
Now, this one is interesting.
Oops.
Again I did it.
I forgot to write whether it's simply connected or not.
Let me come back over to four for posterity.
Not simply connected.
OK, sorry about that.
For the fifth one, because I'm in two dimensions, it's going
to be not simply connected, but if I add a third
dimension, it would become simply connected.
So I want to explain why it's not simply connected here, and
then I want to show you why in a third dimension it becomes
simply connected.
OK?
The problem curves are the curves that do this, that go
around this line segment.
Because notice, if I want to try and contract this curve
down to a point and I don't want to intersect that line
segment, in order to do it I'd actually have to move it away
from the line segment.
I'd have to pass through the line segment.
At some point, this curve would intersect that segment
in order to be able to shrink it to a point in the region
I'm interested in.
So this segment is getting in the way-- we can think of it
that way-- of allowing me to contract this down to a point.
Actually also, when we talked about simply connectedness in
two dimensions, it was easier.
Because we could say, if we take any curve and we look at
the disk that's spanned by the boundary of this curve, and we
look at the region the curve encloses, notice that this
segment is in that region.
And there's no way of drawing this kind of curve without the
segment being in that region, and that's how we know it's
not simply connected.
Now, in three dimensions, what happens?
What if I took this exact same picture and I just made the
z-axis come out from the board?
Why is that suddenly simply connected, whereas in the
two-dimensional case it's not?
And the reason is because in this same picture, I could
take this same curve, and I could take this shaded thing,
and I could push the shaded thing out of the xy plane.
And so I'd still have the same boundary curve, but I'd have
the shaded portion not hitting the segment.
And so I can find some surface with this boundary that
doesn't have this segment in the interior of the surface.
And that's another way of thinking about simply
connectedness.
So in the two-dimensional case, it is not simply
connected, but if I were to add a third dimension, this
region would become simply connected.
OK.
Because I would have no problem for any curve finding
some surface that had that curve as a boundary that
didn't intersect that segment.
So I could keep the surface in the region I
was interested in.
OK.
So that would tell me it was simply connected.
And then the last one is a solid torus.
OK, and this one, we might not have dealt with solid tori
before, but this is an interesting problem.
OK, so there are fundamentally--
we say in math-- that there are two classes of curves that
are interesting.
We won't get into the exact terminology of what's
happening, but there are two types of curves on the torus.
One type of curve is the kind that goes around right here.
OK.
So it loops around the doughnut in that direction.
But that type of curve is nice, because notice, that if
I look at the surface in there, it's all inside the
solid torus.
So that's good.
So that seems like that's a curve that is not telling us
it's not simply connected.
We'll say that.
But there's another class of curves in the torus.
And that's the class of curves that goes around--
this is a little harder to draw--
the top, but around the hole.
OK?
Around the hole.
Now any surface I have that I try to draw--
any surface that's going to have
that curve as a boundary--
is at some point forced to leave the solid torus.
And the reason is really because of
the hole in the middle.
Right?
That's really the reason it happens.
OK.
And so you can see the part right in here is on the
surface, but it's not in the solid torus.
So because I have a curve that any surface I draw that has
that curve as a boundary is forced to leave the solid
torus, it's a non-simply connected region.
So we say not simply connected.
OK.
So I'm going to go back through real quickly and just
remind us what was happening.
And maybe use the language I was using at the end to
describe the first examples, because that might help a
little better.
So let's go back to the first examples.
OK, in the R3 example, again, number one, we know it's
simply connected.
We're not going to worry about it.
OK.
But let me draw number two.
Maybe if I draw some shaded region, this will help us
understand it a little bit better.
Number two we established was not simply connected.
And if you think about it, if you have a curve that goes
around the z-axis, and you want to look at a surface that
has that curve as its boundary, this surface
certainly intersects the z-axis.
The question is, can I keep this curve the way it is, and
pull the surface away and have it not intersect the z-axis?
And the answer is no.
Any way I move the inside of the curve-- basically, what
looks like a disk--
it's still going to intersect the z-axis somewhere.
Right?
And so it's definitely not simply connected.
And the thing I was trying to point out in number three is
that it is simply connected.
If I shade the boundary of a curve sitting in the xy plane,
and then I take that shaded disk and I push it up a
little, then it no longer hits the origin.
And I haven't fundamentally changed my curve at all.
And so that's a way of understanding that it is
actually simply connected.
OK?
So there are a couple of ways to think about it.
And without being incredibly mathematically precise, these
are some of the best ways we have of thinking about
understanding simply connected or not simply connected.
So again, we had six examples.
Removing the z-axis from R3 was not simply connected.
Removing the origin from R3 was still simply connected.
Removing a circle from R3 was not simply connected for the
same reason as the z-axis problem, because here was my
disk, and any way I try to move this shaded surface, I
can't keep it from intersecting this circle.
And then number five was R2 minus a segment.
It was not simply connected, but if I add another
dimension, it is simply connected, for the same kind
of reason that R3 minus the origin was.
And then number six was the solid torus.
Which now, it's kind of hard to see what the solid torus
looks like.
But we said, there's one curve that behaves fine, but the
curve that goes all the way around the hole shows it's, in
fact, not simply connected.
So hopefully that was informative, and
that's where I'll stop.