# Compact n-manifolds as Compactifications of R^n

• Bacle
In summary: I can see, e.g., for n=2, we can construct a manifold by using an n-gon, and identifying some edges. Then the interior appears clearly to be homeo. to R^n. In some other cases, e.g., S^n, this seems easy to show using the stereo projection, and some work with the topology defined on the (Alexandroff) 1-pt. compactification ( use as open sets in S^n, all open sets in C --or in R^n --together with complements of compact subsets K of C.) Then R^n is
Bacle
Hi, I am trying to show that RP^n is the
compactification of R^n. I have some , but not all I
need:

I have also heard the claim that every compact n-manifold is
a compactification of R^n, but I cannot find a good general
argument . I can see, e.g., for n=2, we can construct a manifold
by using an n-gon, and identifying some edges. Then the interior
seems clearly to be homeo. to R^n.

In some other cases, e.g., S^n, this seems easy to show
using the stereo projection, and some work with the topology
defined on the (Alexandroff) 1-pt. compactification ( use
as open sets in S^n, all open sets in C --or in R^n --together
with complements of compact subsets K of C.) Then R^n is open
in S^n, and its closure is necessarily S^n itself, i.e., R^n is
embedded densely in the compact space S^n ( S^n is
compact by construction.)

For the sake of practicing my pointset topology and some
basic diff. geometry, I tried to show that RP^n is also a
compactification of R^n, before trying a more general argument:

i) I know RP^n is compact, as the quotient of the
compact space S^n by a continuous (quotient) map.

ii) I think I can show R^n is dense in RP^n, since
the identification of S^n is made only in the
boundary of S^n, and the interior of S^n is homeo.
to R^n, i.e., missing some boundary points. The
interior of the disk (homeo. to R^n ) is dense in the
disk. Is the image
(under the quotient map) also dense in RP^n?

Do I need something else?

Let me be more specific:
The quotient map
q:S^n-->S^n/~ is continuous, by construction. Then
the continuous image of S^n is compact in the quotient
topology. Right?

Maybe more rigorously, we should start by giving
S^n the subspace topology of R^(n+1). Then, as a closed
subset/subspace of a compact space, it is
itself compact.

Now: How you have R^n in RP^n (i.e. which embedding) and
why it is
dense in the quotient topology (without hand waving)?

Let's see: if we consider RP^n as the set of
points in S^n/~ ( x~y iff x=ty ; t non-zero ),
then R^n would be everything except for the equator
D^n_ /\D_n+ , i.e., in a "standard" coordinate system
X1,..,Xn, R^n={x=(x1,...,xn);||x||=1 and xn>0 }

Then R^n is a saturated open subset of S^n , under ~
(i.e., the map q(x)=x/~ )so that (since quotient maps
send saturated open sets to open sets) the image
p(R^n) is open in S^n/~

But Cl(R^n)={x=(x1,..,xn);||x||=1, xn>=0 }
is a saturated closed set re q(x)=x/~. So
q(Cl(R^n)) is closed in RP^n.

Now we need to show that Cl(q(R^n))=RP^n

But for any point z in q(Cl(R^n)), where
xn=0 for z, we have:

q^-1( q(Cl(R^n))-{z} ) is not closed in
(S^n, subspace) (though I don't have a good
argument to this effect)

Then q((Cl(R^n)) =RP^n is the closure of
(the embedded image of ) R^n in RP^n, and
so R^n is dense in RP^n.

This seems quite good. However, compactifications are usually required to be Hausdorff. Non-Hausdorff compactifications are quite useless in my opinion. So it would be good if you also showed that RP^n is Hausdorff. But I guess you already showed this by showing that RP^n is a manifold?

Thanks, Micromass, for going through that messy post:

To show Hausdorff, we could take, given any x,y in RP^n , q-saturated

'hoods (neighborhoods) U_x, U_y in S^n (quotient maps send saturated open

sets to open sets ), of arc-length less than Pi each, and such that the total length

of the union is less than Pi , so that there are no antipodal pairs in U_x\/U_y.

Then , for q(x)=x/`~ , the natural projection, O_x=q(U_x) and O_y= q(U_y) are open

in RP^n. And I claim they are disjoint too: if there was a [z] in the overlap

then either:

i) there is a z in S^1 common to both U_x, U_y. Not possible by construction

ii) There is a z in U_x , and its antipode -z in U_y . Also not possible by construction.

Still, I am looking for a general argument for why a compact n-manifold --and therefore

Hausdorff-- is a compactification of R^n

The covering map of the sphere onto projective space is continuous and onto. Since the sphere is compact, this implies that projective space is also compact.

A fundamental domain of the covering map is any closed hemisphere. A closed hemisphere is homeomorphic to an n-ball.

I think that your problem is more accurately stated by saying that a compact manifold is always obtained from a close n-ball via identifications on its boundary n-1 sphere. I got a little confused when you said that is was a compactification of Euclidean space.

This theorem, if it is true, I would think could be proved by following the exponential mapping at some point out along geodesics until you obtain a maximal n-ball i.e. an open n ball in the tangent space that is mapped diffeomorphically into the manifold and which can not be enlarged and still be a diffeomorphism. For instance for the sphere you get a ball around the origin in the tangent space whose boundary is identified to a single point.

Last edited:
Lavinia said, in part:

"I think that your problem is more accurately stated by saying that a compact manifold is always obtained from a close n-ball via identifications on its boundary n-1 sphere. I got a little confused when you said that is was a compactification of Euclidean space."

By a compactification K of Euclidean space R^n, I mean that X is a compact manifold,
and that R^n is densely-embedded in K, i.e., that there is an embedding e: R^n-->K
with Cl(e(R^n))=K, with Cl the closure operator.

And an alternative, more topological argument, may be possible if we just consider
compact triangulatable ( I don't know if this is an actual word) manifolds, and then we
can check a finite number of triangles/simplices.

But, yes, if we could show that every compact n-manifold can be obtained by
identification, that would do it, but I did not know how to generalize the process
of identifying the edges of fundamental surfaces, nor how to prove that every
compact manifold K could be obtained that way.

And your idea of the exponential map sounds good, thanks, I will try it out.

## 1. What is a compact n-manifold?

A compact n-manifold is a mathematical object that can be thought of as a generalization of a surface in three-dimensional space. It is a topological space that is locally homeomorphic to n-dimensional Euclidean space, but is also compact, meaning it is both closed and bounded.

## 2. How is a compact n-manifold different from a non-compact one?

The main difference between a compact n-manifold and a non-compact one is that a compact n-manifold has a finite size and is bounded, while a non-compact one can be infinite in size and unbounded. This has important implications for the behavior and properties of the manifold.

## 3. What is the significance of compact n-manifolds as compactifications of R^n?

Compact n-manifolds serve as compactifications of n-dimensional Euclidean space, meaning they can be thought of as a way to "enclose" or "wrap up" n-dimensional space in a compact and finite way. This has important applications in different areas of mathematics, such as topology and differential geometry.

## 4. How are compact n-manifolds used in physics?

Compact n-manifolds are used in physics to model and study various physical phenomena, such as the behavior of particles in space or the geometry of spacetime. They can also be used in string theory and other areas of theoretical physics to describe the structure of the universe.

## 5. What are some examples of compact n-manifolds?

Some examples of compact n-manifolds include the n-sphere, n-torus, n-dimensional projective space, and any closed and bounded subset of n-dimensional Euclidean space. These manifolds have important applications in mathematics, physics, and other fields.

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