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 R^3.
The second one is if I take R^3
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
R^3 and remove a circle.
The fifth one is R^2
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 R^3 is simply connected.
But if you weren't sure
about it, you could think,
any closed curve I draw in
R^3, I can certainly get all
of the inside of it
contained in R^3.
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 R^3.
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 R^3.
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-- you know,
if 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
R^3 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-- there's
a big open circle at the origin.
That's not included in
our domain, in our region.
So our region is all of
R^3 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
R^3 minus a circle.
So let me see if I can
draw a picture of that.
And 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 R^2
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.
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 this curve-- where
the boundary is 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
promotes simply connectedness.
Or it's 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, but say around 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 R^3 example,
again, number one,
we know it's simply connected.
We're not going
to worry about it.
OK.
But let me draw--
in 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,
that it is simply
connected, is 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 R^3
was not simply connected.
Removing the origin from R^3
was still simply connected.
Removing a circle from R^3
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 R^2 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 R^3
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 kind
of 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.
