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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 understanding

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simply connected regions
in three dimensions.

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Well, there's one
two-dimensional one,

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but the rest are
three dimensions.

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So what I want you
to do is for each

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of the following-- there are six
different regions-- determine

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whether or not each of
them is simply connected.

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So the first one is R^3.

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The second one is if I take R^3
and I 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
R^3 and remove a circle.

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The fifth one is R^2
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,

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and it includes the
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,

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is determine whether or
not each of these regions

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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

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ready to check your work.

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OK, welcome back.

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So again, what we're
interested in doing

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is understanding
simply connectedness

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in another dimension.

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We did something
already, a while back,

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with two dimensions,
and so now we

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want to understand it
better in 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 R^3 is simply connected.

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

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any closed curve I draw in
R^3, I can certainly get all

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of the inside of it
contained in R^3.

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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 R^3.

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So then the first one is an easy
yes to simply connectedness.

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OK?

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So let's start on
the second one,

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and I'm going to draw a
little picture for us.

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So the second one is R^3.

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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

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so we know we're removing
that part from R3.

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And I'm removing
it all the way up

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to minus infinity
in the z-direction

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and plus infinity
in the z-direction.

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

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that when I try and compress
that closed curve down

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to a point, I can't
do it while remaining

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inside this region that is
all of R3 minus the z-axis.

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And the answer is
there is a whole family

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of curves that do this.

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

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you'll notice that there's
something on the inside of it,

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regardless of-- you know,
if I slide it up or down,

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there's a point on the
inside of this curve that

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is not in the region
I'm interested in.

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The region, again, is
R^3 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

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and I were to put a
surface across this curve,

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so it was like a
disk, there would

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be a point on the z-axis
that would intersect it.

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

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

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

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the 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,

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I just remove 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-- there's

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a big open circle at the origin.

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That's not included in
our domain, in our region.

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So our region is all of
R^3 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

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than what you had previously.

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

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even if I take a curve that's
sitting in the xy-plane that

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goes around the
origin, the point

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is I can keep this curve
in three-dimensional space,

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and I can wiggle it around,
so that I can shrink it down

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to a point, and the origin
doesn't 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 space

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it was not simply connected, in
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,

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if I drew a curve on the
xy-plane around the origin,

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and I wanted to squish it
down to a point, the only way

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to do that would be to
bring the curve somehow

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through the origin.

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Right?

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I would be stuck having to pass
the curve through the origin

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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,

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I can just kind of
lift it a little bit

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away from the origin, and then
I can shrink it down to a point

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without the origin
getting in the way.

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So having that extra
dimension means

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even though I remove
one point, it's

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still actually a simply
connected region.

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So maybe this is
the first place we

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see that in the
three dimensions we

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have a different case than
we had in two dimensions,

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removing the same
kind of object.

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

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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
R^3 minus a circle.

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

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And 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

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is not simply connected, because
of one particular problem.

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It's actually the
same kind of problem

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you have when you
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-- notice

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that any way I try
and move this curve

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and shrink it down to
a point, this circle

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is going to get in the
way for the same reason

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that the z-axis got in the way.

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Because this circle
is closed, I can't

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slide the curve I'm interested
in away from the circle

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and 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

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where we just had the origin,
because we could take any curve

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and we could move it
away from the origin,

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and then 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,

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it's going to have to
hit the circle if I

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want to move it away
so I can shrink it

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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 R^2
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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The fifth one, because
I'm in two dimensions,

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it's going to be not
simply connected,

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but if I add a
third dimension, it

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would become simply connected.

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

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and then I want to show you
why in a third dimension

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it becomes simply connected.

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OK?

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The problem curves
are the curves

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that do this, that go
around this line segment.

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

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

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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 I'm

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interested in.

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So this segment is
getting in the way--

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we can think of it that way--
of allowing me to contract this

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down to a point.

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Actually also, when we talked
about simply connectedness

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in two dimensions,
it was easier.

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Because we could say,
if we take any curve

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and we look at the disk that's
spanned by this curve-- where

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the boundary is this curve,
and we look at the region

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the curve encloses-- notice that
this segment is in that region.

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And there's no way of
drawing this kind of curve

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without the segment
being in that region,

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and that's how we know
it's 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

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and I just made the z-axis
come out from the board?

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Why is that suddenly
simply connected,

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whereas in the
two-dimensional case it's not?

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And the reason is because
in this same picture,

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I could take this
same curve, and I

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could take this
shaded thing, and I

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could push the shaded
thing out of the xy-plane.

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

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but I'd have the shaded portion
not hitting the segment.

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And so I can find some
surface with this boundary

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

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And that's another
way of thinking

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about simply connectedness.

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

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but if I were to add
a third dimension,

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this 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

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that didn't intersect
that segment.

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So I could keep the surface in
the region I 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-- we

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

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that 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,

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that if I look at
the surface in there,

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it's all inside the solid torus.

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So that's good.

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So that seems like
that's a curve that

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promotes simply connectedness.

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Or it's not telling us
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, but say around the top,

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but around the hole.

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OK?

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Around the hole.

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00:11:16,640 --> 00:11:19,630
Now any surface
I have that I try

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00:11:19,630 --> 00:11:24,880
to draw-- any
surface that's going

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00:11:24,880 --> 00:11:27,210
to have that curve
as a boundary-- is

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at some point forced to
leave the solid torus.

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And the reason is really because
of the hole in the middle.

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Right?

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00:11:33,780 --> 00:11:36,170
That's really the
reason it happens.

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00:11:36,170 --> 00:11:37,270
OK.

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00:11:37,270 --> 00:11:41,140
And so you can see
the part right in here

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00:11:41,140 --> 00:11:45,820
is on the surface, but it's
not in the solid torus.

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00:11:45,820 --> 00:11:48,780
So because I have a
curve that any surface

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00:11:48,780 --> 00:11:51,410
I draw that has that
curve as a boundary is

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00:11:51,410 --> 00:11:54,290
forced to leave the
solid torus, it's

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00:11:54,290 --> 00:11:56,540
a non-simply-connected region.

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00:11:56,540 --> 00:11:57,870
So we say not simply connected.

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00:12:01,501 --> 00:12:02,000
OK.

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00:12:02,000 --> 00:12:05,120
So I'm going to go back through
real quickly and just remind us

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what was happening.

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And maybe use the language
I was using at the end

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00:12:10,107 --> 00:12:11,940
to describe the first
examples, because that

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00:12:11,940 --> 00:12:13,064
might help a little better.

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00:12:13,064 --> 00:12:15,290
So let's go back to
the first examples.

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00:12:15,290 --> 00:12:17,730
OK, in the R^3 example,
again, number one,

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we know it's simply connected.

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00:12:19,610 --> 00:12:21,420
We're not going
to worry about it.

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OK.

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00:12:22,000 --> 00:12:24,760
But let me draw--
in number two, maybe

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if I draw some
shaded region, this

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00:12:26,312 --> 00:12:28,270
will help us understand
it a little bit better.

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Number two we established
was not simply connected.

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00:12:30,790 --> 00:12:32,550
And if you think
about it, if you

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have a curve that goes
around the z-axis,

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00:12:34,390 --> 00:12:36,540
and you want to look
at a surface that

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has that curve as its boundary,
this surface certainly

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00:12:40,370 --> 00:12:41,504
intersects the z-axis.

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00:12:41,504 --> 00:12:43,420
The question is, can I
keep this curve the way

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00:12:43,420 --> 00:12:47,380
it is, and pull the surface
away and have it not

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00:12:47,380 --> 00:12:48,910
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 --> 00:12:54,430
Any way I move the
inside of the curve--

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00:12:54,430 --> 00:12:56,890
basically, what looks
like a disk-- it's still

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00:12:56,890 --> 00:12:59,120
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,

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00:13:06,330 --> 00:13:09,350
that it is simply
connected, is if I

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00:13:09,350 --> 00:13:12,840
shade the boundary of a curve
sitting in the xy-plane,

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00:13:12,840 --> 00:13:16,840
and then I take that shaded
disk and I push it up a little,

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00:13:16,840 --> 00:13:20,110
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 --> 00:13:25,680
And so that's a way of
understanding that it

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00:13:25,680 --> 00:13:27,470
is actually simply connected.

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00:13:27,470 --> 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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00:13:31,810 --> 00:13:35,194
And without being incredibly
mathematically precise,

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00:13:35,194 --> 00:13:36,610
these are some of
the best ways we

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00:13:36,610 --> 00:13:40,220
have of thinking about
understanding simply connected

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00:13:40,220 --> 00:13:41,930
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 R^3
was not simply connected.

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00:13:48,110 --> 00:13:53,120
Removing the origin from R^3
was still simply connected.

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00:13:53,120 --> 00:13:56,910
Removing a circle from R^3
was not simply connected

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00:13:56,910 --> 00:14:00,720
for the same reason
as the z-axis problem,

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00:14:00,720 --> 00:14:06,000
because here was my disk, and
any way I try to move this

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00:14:06,000 --> 00:14:12,240
shaded surface, I 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 R^2 minus a segment.

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00:14:15,946 --> 00:14:18,975
It was not simply connected,
but if I add another dimension,

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00:14:18,975 --> 00:14:22,090
it is simply connected, for the
same kind of reason that R^3

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00:14:22,090 --> 00:14:24,110
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 --> 00:14:28,060
Which now, it's kind
of hard to see what

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00:14:28,060 --> 00:14:30,920
the solid torus looks like.

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00:14:30,920 --> 00:14:35,230
But we said, there's one kind
of curve that behaves fine,

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00:14:35,230 --> 00:14:38,030
but the curve that goes
all the way around the hole

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00:14:38,030 --> 00:14:41,360
shows it's, in fact,
not simply connected.

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00:14:41,360 --> 00:14:43,350
So hopefully that
was informative,

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00:14:43,350 --> 00:14:45,140
and that's where I'll stop.