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CHRISTINE BREINER: Welcome
back to recitation.

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In this video, what I'd like
us to do is work on

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understanding simply connected
regions in three dimensions.

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Well, there's one
two-dimensional one, but the

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rest are three dimensions.

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So what I want you to do is for
each of the following--

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there are six different
regions--

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determine whether or not each
of them is simply connected.

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So the first one is R3.

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The second one is if
I take R3 and I

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remove the entire z-axis.

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The third one is if I take
R3 and I remove 0.

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The fourth one is if I take
R3 and remove a circle.

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The fifth one is R2 minus
a line segment.

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And the sixth one is
a solid torus.

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So a solid torus looks like a
doughnut, and it includes the

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inside of the doughnut.

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This looks like a doughnut,
hopefully, to you.

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And it's not hollow.

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It includes the inside.

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So what I'd like you to do,
again, is determine whether or

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not each of these regions
is simply connected.

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And why don't you pause the
video while you work on that.

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And then bring the video back
up when you're ready

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to check your work.

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OK, welcome back.

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So again, what we're interested
in doing is

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understanding simply

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connectedness in another dimension.

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We did something already
a while back with two

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dimensions, and so now we want
to understand it better in

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three dimensions.

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So let's work through these.

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Well, I'm not going to write
anything down for number one,

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because you should
already know that

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R3 is simply connected.

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But if you weren't sure about
it, you could think, for any

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closed curve I draw in R3, I can
certainly get all of the

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inside of it contained in R3.

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Another way to think about it is
that I can take that curve

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and I can collapse it down to
a point, and remain in R3.

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So then the first one is
an easy yes to simply

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connectedness.

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OK?

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So let's start on the second
one, and I'm going to draw a

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little picture for us.

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So the second one is R3.

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I should go this way.

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This is x, y, and z, but then
I remove the entire z-axis.

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So I should make this really
dark so we know we're removing

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that part from R3.

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And I'm removing it all the way
up to minus infinity in

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the z-direction and plus
infinity in the z-direction.

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Now, the question is can I find
any closed curve, that

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when I try and compress that
closed curve down to a point,

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I can't do it while remaining
inside this region that is all

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of R3 minus the z-axis.

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And the answer is there
is a whole family of

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curves that do this.

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If I take a curve that goes
around the z-axis, you'll

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notice that there's something
on the inside of it--

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regardless, of whether I slide
it up or down-- there's a

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point on the inside of this
curve that is not in the

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region I'm interested in.

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The region, again, is
R3 minus the z-axis.

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So there are two ways
to think about this.

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You can think about if I were to
take this curve and I were

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to put a surface across this
curve so it was like a disk,

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there would be a point
on the z-axis that

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would intersect it.

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Or you can think about it as
saying, I have this curve and

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if I try and squeeze it down to
as small as I can get it, I

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can't get it as small is I
want without hitting the

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z-axis at some point.

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The z-axis is kind of
in the way, right?

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Now, number three is a little
different situation.

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Because in number three, I think
this exact same picture,

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but instead of removing the
whole z-axis, I just remove

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the origin.

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So let me try and draw
a picture of that.

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So I'm going to make this a big
open circle at the origin.

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That's not included
in our region.

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So our region is all of
R3 except the origin.

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And in two-dimensional space,
this was not simply connected.

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But in three-dimensional space
it is simply connected.

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So this is a little different
situation than what you had

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previously.

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And so the idea is here, if I
take a curve, even if I take a

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curve that's sitting in the xy
plane that goes around the

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origin, the point is I can
keep this curve in

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three-dimensional space, and I
can wiggle it around, so that

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I can shrink it down to a point,
and the origin doesn't

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get in the way.

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It doesn't keep me
from doing that.

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So actually, this region, even
though in two-dimensional

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space it was not simply
connected, in

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three-dimensional space it is.

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And let's see if we understand
the difference.

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The difference is in
two-dimensional space, if I

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drew a curve on the xy plane
around the origin, and I

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wanted to squish it down to a
point, the only way to do that

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would be to bring the curve
somehow through the origin.

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Right?

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I would be stuck having to pass
the curve through the

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origin to shrink it
down to a point.

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But in three space, I have
another dimension.

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So a curve that sits on the xy
plane, I can just kind of lift

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it a little bit away from the
origin, and then I can shrink

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it down to a point without the
origin getting in the way.

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So having that extra dimension
means even though I remove one

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point, it's still actually a
simply connected region.

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So maybe this is the first place
we see that in the three

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dimensions we have a different
case than we had in two

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dimensions removing the
same kind of object.

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So I realize now I haven't
been writing down whether

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these are simply connected
or not.

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So I should write down this
is simply connected.

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And maybe for number two I
should go back and formally

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write not simply connected.

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So that we have this
for posterity.

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Now the fourth one is
R3 minus the circle.

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So let me see if I can draw
a picture of that.

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And for the circle, it doesn't
really matter where it is.

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I'm just going to draw
one somewhere.

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So here's my circle.

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So everything is in my region
except this circle.

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And the question: is it
simply connected?

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And the answer is: no, the
region is not simply

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connected, because of one
particular problem.

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It's actually the same kind of
problem you have when you

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remove the z-axis.

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And that is, if I draw a curve
that goes around this circle--

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any curve that goes around
this circle--

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notice that any way I try and
move this curve and shrink it

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down to a point, this circle is
going to get in the way for

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the same reason that the
z-axis got in the way.

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Because this circle is closed,
I can't slide the curve I'm

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interested in away from
the circle and

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then shrink it down.

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OK.

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There's some sort of obstruction
right here.

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And so it's fundamentally
different than the case where

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we just had the origin, because
we could take any

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curve and we could move it away
from the origin, and then

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shrink it down to a point.

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And the origin didn't
get in the way.

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But here, anywhere I try and
move this curve, it's going to

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have to hit the circle if I want
to move it away so I can

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shrink it to a point
in my region.

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So this circle is preventing
me from shrinking it down.

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OK, and then there
are two more.

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And the fifth one is R2
minus a line segment.

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So now we're in two-dimensional
space.

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OK, and let me just
pick a segment.

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OK.

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Now, this one is interesting.

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Oops.

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Again I did it.

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I forgot to write whether it's
simply connected or not.

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Let me come back over to
four for posterity.

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Not simply connected.

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OK, sorry about that.

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For the fifth one, because I'm
in two dimensions, it's going

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to be not simply connected,
but if I add a third

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dimension, it would become
simply connected.

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So I want to explain why it's
not simply connected here, and

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then I want to show you why in
a third dimension it becomes

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simply connected.

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OK?

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The problem curves are the
curves that do this, that go

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around this line segment.

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Because notice, if I want to
try and contract this curve

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down to a point and I don't want
to intersect that line

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segment, in order to do it I'd
actually have to move it away

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from the line segment.

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I'd have to pass through
the line segment.

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At some point, this curve would
intersect that segment

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in order to be able to shrink
it to a point in the region

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I'm interested in.

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So this segment is getting in
the way-- we can think of it

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that way-- of allowing me to
contract this down to a point.

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Actually also, when we talked
about simply connectedness in

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two dimensions, it was easier.

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Because we could say, if we take
any curve and we look at

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the disk that's spanned by the
boundary of this curve, and we

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look at the region the curve
encloses, notice that this

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segment is in that region.

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And there's no way of drawing
this kind of curve without the

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segment being in that region,
and that's how we know it's

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not simply connected.

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Now, in three dimensions,
what happens?

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What if I took this exact same
picture and I just made the

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z-axis come out from
the board?

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Why is that suddenly simply
connected, whereas in the

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two-dimensional case it's not?

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And the reason is because in
this same picture, I could

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take this same curve, and I
could take this shaded thing,

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and I could push the shaded
thing out of the xy plane.

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And so I'd still have the same
boundary curve, but I'd have

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the shaded portion not
hitting the segment.

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And so I can find some surface
with this boundary that

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doesn't have this segment in the
interior of the surface.

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And that's another way of
thinking about simply

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connectedness.

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So in the two-dimensional
case, it is not simply

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connected, but if I were to add
a third dimension, this

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region would become
simply connected.

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OK.

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Because I would have no problem
for any curve finding

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some surface that had that
curve as a boundary that

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didn't intersect that segment.

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So I could keep the surface
in the region I

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was interested in.

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OK.

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So that would tell me it
was simply connected.

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And then the last one
is a solid torus.

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OK, and this one, we might not
have dealt with solid tori

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before, but this is an
interesting problem.

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OK, so there are
fundamentally--

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we say in math-- that there are
two classes of curves that

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are interesting.

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We won't get into the exact
terminology of what's

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happening, but there are two
types of curves on the torus.

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One type of curve is the kind
that goes around right here.

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OK.

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So it loops around the doughnut
in that direction.

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But that type of curve is nice,
because notice, that if

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I look at the surface in there,
it's all inside the

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solid torus.

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So that's good.

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So that seems like that's a
curve that is not telling us

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it's not simply connected.

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We'll say that.

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But there's another class
of curves in the torus.

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And that's the class of curves
that goes around--

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this is a little harder
to draw--

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the top, but around the hole.

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OK?

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Around the hole.

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00:11:16,640 --> 00:11:23,740
Now any surface I have
that I try to draw--

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any surface that's
going to have

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that curve as a boundary--

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is at some point forced to
leave the solid torus.

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And the reason is really
because of

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the hole in the middle.

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Right?

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That's really the reason
it happens.

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OK.

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And so you can see the part
right in here is on the

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surface, but it's not
in the solid torus.

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So because I have a curve that
any surface I draw that has

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that curve as a boundary is
forced to leave the solid

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torus, it's a non-simply
connected region.

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So we say not simply
connected.

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OK.

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So I'm going to go back through
real quickly and just

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remind us what was happening.

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And maybe use the language
I was using at the end to

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describe the first examples,
because that might help a

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little better.

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So let's go back to the
first examples.

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OK, in the R3 example, again,
number one, we know it's

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simply connected.

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We're not going to
worry about it.

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OK.

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But let me draw number two.

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Maybe if I draw some shaded
region, this will help us

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00:12:27,090 --> 00:12:28,270
understand it a little
bit better.

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Number two we established was
not simply connected.

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And if you think about it, if
you have a curve that goes

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around the z-axis, and you want
to look at a surface that

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has that curve as its boundary,
this surface

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certainly intersects
the z-axis.

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The question is, can I keep this
curve the way it is, and

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00:12:44,940 --> 00:12:48,910
pull the surface away and have
it not intersect the z-axis?

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00:12:48,910 --> 00:12:52,140
And the answer is no.

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00:12:52,140 --> 00:12:55,150
Any way I move the inside of
the curve-- basically, what

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00:12:55,150 --> 00:12:56,360
looks like a disk--

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00:12:56,360 --> 00:12:59,120
it's still going to intersect
the z-axis somewhere.

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00:12:59,120 --> 00:12:59,710
Right?

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00:12:59,710 --> 00:13:02,140
And so it's definitely
not simply connected.

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00:13:02,140 --> 00:13:06,330
And the thing I was trying to
point out in number three is

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that it is simply connected.

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00:13:08,670 --> 00:13:12,840
If I shade the boundary of a
curve sitting in the xy plane,

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and then I take that shaded
disk and I push it up a

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00:13:15,760 --> 00:13:20,110
little, then it no longer
hits the origin.

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00:13:20,110 --> 00:13:22,970
And I haven't fundamentally
changed my curve at all.

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00:13:22,970 --> 00:13:25,860
And so that's a way of
understanding that it is

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00:13:25,860 --> 00:13:27,730
actually simply connected.

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00:13:27,730 --> 00:13:27,970
OK?

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00:13:27,970 --> 00:13:31,810
So there are a couple of
ways to think about it.

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And without being incredibly
mathematically precise, these

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00:13:35,380 --> 00:13:38,640
are some of the best ways we
have of thinking about

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00:13:38,640 --> 00:13:41,930
understanding simply connected
or not simply connected.

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00:13:41,930 --> 00:13:45,150
So again, we had six examples.

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00:13:45,150 --> 00:13:48,110
Removing the z-axis from R3
was not simply connected.

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00:13:48,110 --> 00:13:53,120
Removing the origin from R3 was
still simply connected.

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00:13:53,120 --> 00:13:57,130
Removing a circle from R3 was
not simply connected for the

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00:13:57,130 --> 00:14:02,740
same reason as the z-axis
problem, because here was my

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00:14:02,740 --> 00:14:08,650
disk, and any way I try to move
this shaded surface, I

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00:14:08,650 --> 00:14:12,240
can't keep it from intersecting
this circle.

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00:14:12,240 --> 00:14:15,946
And then number five was
R2 minus a segment.

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00:14:15,946 --> 00:14:18,720
It was not simply connected,
but if I add another

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00:14:18,720 --> 00:14:21,250
dimension, it is simply
connected, for the same kind

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00:14:21,250 --> 00:14:24,110
of reason that R3 minus
the origin was.

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00:14:24,110 --> 00:14:26,120
And then number six was
the solid torus.

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00:14:26,120 --> 00:14:28,820
Which now, it's kind of hard
to see what the solid torus

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00:14:28,820 --> 00:14:30,920
looks like.

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00:14:30,920 --> 00:14:35,520
But we said, there's one curve
that behaves fine, but the

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00:14:35,520 --> 00:14:38,630
curve that goes all the way
around the hole shows it's, in

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00:14:38,630 --> 00:14:41,360
fact, not simply connected.

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00:14:41,360 --> 00:14:43,490
So hopefully that was
informative, and

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00:14:43,490 --> 00:14:45,140
that's where I'll stop.

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00:14:45,140 --> 00:14:46,219