the Bohr compactification of the Reals

**marcus** · Feb16-04, 12:45 PM

the Bohr compactification of the real line
R_Bohr
is essential (it seems) to Loop Quantum Cosmology.

Here are some links illustrating that:
http://www.physicsforums.com/showthr...432#post147432

I want to understand the Bohr compactification better.

It is putting a different topology on the Reals
(or the reals embedded as a dense subset of a larger
collection of numbers) so that they become compact.
But it is not the "one point" compactification which
is a widely known trick of adjoining a "point at infinity".
It is a different and probably more cool compactification.
The younger brother of Niels thought of it. His name was Harald.

Well, why shouldnt we just keep on using the line of real numbers
that we know and love, but have a special topology handy for
use on Sundays and other special occasions that makes them compact.
Sounds OK to me.

To understand the construction we probably need to know the idea of the "dual group" or (a droll synonym) the "group of characters" of a commutative group G.
The characters of a commut. topol. group G are just the continuous homomorphisms from G to the Unit Circle---the complex numbers with |z|=1, under multiplication.
$LaTeX Code: \\chi : G \\longrightarrow S^1$

$LaTeX Code: \\chi(gh) = \\chi(g)\\chi(h)$

a familiar example would be χ(x) = exp(ix)

notice (please) that the characters form a group themselves
you can multiply two of them and it is still continuous
and it still satisfies that multiplicativity condition

I'll post this and get back to it later.

Oh, the bohr cmptfn of the Reals is the dual group of the Reals equipped with a wacko topology called the "discrete" topology in which the real line is totally atomized as if by a giant sneeze and no point is near any other point.

Viqar Husain and Oliver Winkler
"On singularity resolution in quantum gravity"
http://arxiv.org/gr-qc/0312094

Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski
"Mathematical Structure of Loop Quantum Cosmology"
http://arxiv.org/gr-qc/0304074

I see the Ashtekar/Bojowald/Lewandowski paper came out
in the 2003 edition of "Advances in Theoretical and Mathematical Physics" 7, 233-268

**marcus** · Feb18-04, 04:20 PM

Wikipedia
http://en.wikipedia.org/wiki/Pontrya...st-periodicity

the first reference the article gives is Walter Rudin
Fourier Analysis on Groups (Interscience 1962)

Bohr compactification and almost-periodicity

One use made of Pontryagin duality is to give a general definition of an almost-periodic function on a non-compact group G in LCA. For that, we define the Bohr compactification B(G) of G as H', where H is as a group G', but given the discrete topology. Since H -> G' is continuous and a homomorphism, the dual morphism G -> B(G) is defined, and realizes G as a subgroup of a compact group. The restriction to G of continuous functions on B(G) gives a class of almost-periodic functions; one can imagine them as analogous to the restrictions to a copy of R wound round a torus.

the idea is that duality is reflexive so you expect G'' = G
the dual of the dual of a group G is the same group back again
But suppose you take the dual of G and throw away the usual topology and give it the discrete topology (the power set, same as saying that singletons are open)
so you say H is equal to G' as a group but atomized as a topological space and all functions from it are continuous.
THEN you take the dual of H, and that is the bohr compactification of G.

so it is almost like taking G'' and getting G back
except you intervene in midstream and change the topology.

**marcus** · Feb18-04, 04:50 PM

Now what Wikipedia means by the map H->G'
is the identity map and of course it is
continuous (as Wiki notes) because it is coming from the discrete topology from whence all maps are continuous.

so the identity map H-> G' is a continuous homomorphism and Wiki
says look at the "dual morphism" to that
(morphisms come in pairs like Mounds bars, the chocolate coated
cocoanut treat)

**marcus** · Feb18-04, 04:56 PM

we are going to "lift" the identity map to its dual
B(G) is the dual of H, so put it upstairs over H

and G'' is the same as G so G is the dual of G'
and lets put G upstairs over G'
and draw this little mapping diagram such as the people
in the chalk-dust clouds on the third floor of the math building like to draw.

$LaTeX Code: B(G) \\leftarrow G$
$LaTeX Code: H \\rightarrow Gsingle-quote$

notice that the dual morphism arrow points backwards

**marcus** · Feb18-04, 08:27 PM

to "lift" the identity map to its dual morphism
(what Wiki calls it)
I have to say what g, an element of G, gets mapped to

it has to be mapped to something in B(G)
which is defined as the dual of H

you think i get a kick out of this?
it is trivial
g gets mapped to the simple act of evaluating
a χ at g.

remember H consists of homomorphisms χ from G to the unit circle S.
$LaTeX Code: \\chi : G \\rightarrow S ; g \\rightarrow \\chi (g)$

so a group element g can actually behave as a homomorphism from H to the unit circle

$LaTeX Code: g : H \\rightarrow S ; \\chi \\rightarrow \\chi (g)$

$LaTeX Code: B(G) \\leftarrow G$
$LaTeX Code: H \\rightarrow Gsingle-quote$

you are supposed to check that it is in fact a homomorphism but
the adept graduate student will just drop a ringing name like
"Pontryagin" and not trouble him or her self with this.

now we begin to see why the discrete topology was used, because
otherwise something might not be continuous and therefore would
be unkosher.

**marcus** · Feb18-04, 08:31 PM

http://en.wikipedia.org/wiki/Pontryagin_duality

"the duality interchanges the categories of discrete groups and compact groups"

OK this must be the meat
we blew the real line to bits and gave it discrete topology just so that we can come home to a dual version of the real line that is compact

here is what Wiki says:

Abstract point of view

In detail, the dual group construction of G^ is a contravariant functor LCA -> LCA^op allowing us to identify the category LCA of locally compact abelian topological groups with its own opposite category. We have G^^ isomorphic to G, in a natural way that is comparable to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces).

The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left R-module, the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = Z so this is true also of the latter.

I am still having difficulty imagining what the bohr compactification of the real line is like. can anyone provide some intuition, imagery, anything helpful to understanding it.
how can the real line (locally compact and abelian but not compact) be mapped into a topological group that is somehow like the real line but compact?

**marcus** · Feb18-04, 08:47 PM

the motivation for this is worth recalling
when general relativity is quantized either quantizing
the metric (geometrodynamics a la Wheeler/Dewitt) or
quantizing the connection (a la Rovelli/Ashtekar/Smolin)
and applied to cosmology then when they use the bohr
compactification to build the quantum state space
then the big bang singularity goes away.
somehow it is a ticket to smoothness at time zero.

Viqar Husain and Oliver Winkler
"On singularity resolution in quantum gravity"
http://arxiv.org/gr-qc/0312094

Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski
"Mathematical Structure of Loop Quantum Cosmology"
http://arxiv.org/gr-qc/0304074

**Hurkyl** · Feb19-04, 01:17 AM

All right, this is a mess. [:)]

All groups in this post will be additive groups. Some of these theorems I've only sketched the proof for, but I'm fairly confident in them. I've omitted the proofs, but I can fill them in if you like.

If you just want the answer, skip to the bottom. [:)]

(If someone more knowledgable can check my work, it would be great! I'm fairly confident in everything, but I feel like I've glossed over a few details)

Let us denote elements of $LaTeX Code: \\mathbb{R}single-quote$ , the dual group to $LaTeX Code: \\mathbb{R}$ by $LaTeX Code: \\chi_r : \\mathbb{R} \\rightarrow \\mathbb{R} / \\mathbb{Z} : x \\rightarrow rx + \\mathbb{Z}$ , where $LaTeX Code: r \\in \\mathbb{R}$ . In other words, $LaTeX Code: \\chi_r(x) = rx (\\mathrm{mod} 1)$ .

(I've chosen to map into $LaTeX Code: \\mathbb{R} / \\mathbb{Z}$ instead of

so that the dual group will be an additive group)

Now, let's let

denote $LaTeX Code: \\mathbb{R}single-quote$ with the discrete topology.

Theorem 1: If

is a group with the discrete topology, then

is the set of all group homomorphisms $LaTeX Code: \\chi : G \\rightarrow \\mathbb{R} / \\mathbb{Z}$ , and a convegence in

coincides with pointwise convergence in $LaTeX Code: \\mathbb{R} / \\mathbb{Z}$ .

Now, the question is "What do group homomorphisms from $LaTeX Code: \\mathbb{R}$ to $LaTeX Code: \\mathbb{R} / \\mathbb{Z}$ look like?" Allow me to emphasize that these homomorphisms do not need to be continuous.

Lemma 2: If $LaTeX Code: \\varphi : \\mathbb{Q} \\rightarrow \\mathbb{R} / \\mathbb{Z}$ is a group homomorphism, then it may be written as $LaTeX Code: \\varphi : \\mathbb{Q} \\rightarrow \\mathbb{R} / \\mathbb{Z} : x \\rightarrow rx + \\mathbb{Z}$ for some $LaTeX Code: r \\in \\mathbb{R}$ .

Theorem 3: If an additive group

is a vector space over $LaTeX Code: \\mathbb{Q}$ , and if If $LaTeX Code: \\varphi : V \\rightarrow \\mathbb{R} / \\mathbb{Z}$ is a group homomorphism, then it may be written as $LaTeX Code: \\varphi : V \\rightarrow \\mathbb{R} / \\mathbb{Z} : x \\rightarrow vx + \\mathbb{Z}$ where

is a real valued linear functional on the vector space

.

Now, it so happens that $LaTeX Code: \\mathbb{R}$ is a vector space (of uncountably infinite dimension) over $LaTeX Code: \\mathbb{Q}$ , so applying theorems 1 and 3:

The dual group of

, $LaTeX Code: B(\\mathbb{R})$ , can be parametrized by the set of all real valued $LaTeX Code: \\mathbb{Q}$ -linear functions on $LaTeX Code: \\mathbb{R}$ . For any such function

, the corresponding element of $LaTeX Code: B(\\mathbb{R})$ is $LaTeX Code: \\varphi_v : \\mathbb{R} \\rightarrow \\mathbb{R}/\\mathbb{Z} : x \\rightarrow vx + \\mathbb{Z}$ .

Convergence of elements in $LaTeX Code: B(\\mathbb{R})$ is given by pointwise convergence over $LaTeX Code: \\mathbb{R} / \\mathbb{Z}$ .

Notice that the additive group $LaTeX Code: \\mathbb{R}single-quote$ is a subgroup of $LaTeX Code: B(\\mathbb{R})$ ; thus, we can use the isomorphism between $LaTeX Code: \\mathbb{R}$ and $LaTeX Code: \\mathbb{R}single-quote$ to embed $LaTeX Code: \\mathbb{R}$ into $LaTeX Code: B(\\mathbb{R})$ .

Theorem 4: If $LaTeX Code: \\varphi_v = \\varphi_w$ , then

.

So we can simplify the formulation somewhat.

-----------------------------------------------------------------------------

The Bohr compactification of $LaTeX Code: \\mathbb{R}$ , $LaTeX Code: B(\\mathbb{R})$ , is the $LaTeX Code: \\mathbb{Q}$ -vector space of all real valued $LaTeX Code: \\mathbb{Q}$ -linear functions on $LaTeX Code: \\mathbb{R}$ . (Equivalently, the set of all additive group homomorphisms from $LaTeX Code: \\mathbb{R}$ to itself)Convergence in $LaTeX Code: B(\\mathbb{R})$ is pointwise.

An element $LaTeX Code: r \\in \\mathbb{R}$ can be identified with the function $LaTeX Code: \\chi_r : \\mathbb{R} \\rightarrow \\mathbb{R} : x \\rightarrow rx$ in $LaTeX Code: B(\\mathbb{R})$ .

**marcus** · Feb19-04, 01:55 AM

portion about typo deleted since was fixed

Hurkyl thanks for the help, have been wanting
help with bohr compactification

**Hurkyl** · Feb19-04, 01:58 AM

Good catch. I cut-pasted the theorem from lemma 2, didn't fix all the definitions!

The end result seems (comparatively) so simple to anything you've quoted from other sources on it; I wonder why this special case isn't mentioned? Have I just made a horrible blunder?

**marcus** · Feb19-04, 02:00 AM

this is the information i believe i needed most, that the real line is an infinite dimensional vectorspace over the field of rational numbers. never occurred to me to think of it that way.
any favorite sourcebook about this?

whoah! sounds from what you just posted that this is a Hurkyl proof!
bravo!
so not cribbed from Walter Rudin or some such venerable

now its after 11PM here so I will not inspect the proof
any more until tomorrow. It seems all right.
and also a good thing to use R/Z instead of unit circle

**Hurkyl** · Feb19-04, 02:11 AM

Erm... Wikipedia? The only mentions of "Bohr compactification" I've seen have actually came from you or sources linked by you. Never understood it until I poured through that Wikipedia link. It was a fun exercise, IMHO. [:D] And I got to learn just what a Haar measure is too, which is nice.

I'm trying to decide if I should come up with a proof that the set I mentioned is, indeed, compact, before I go to bed. I think bed is winning, but I'll know I'll think about it while trying to go to sleep! [:)]

I had seen the reals as a rational vector space before, but just as a curiousity that was introduced by my professors. It's a terrible thing to try and wrap your brain around! [o)] Countably infinite dimensional vector spaces are bad enough!

**Hurkyl** · Feb19-04, 02:40 AM

Grr, Lemma 2 is wrong, methinks, and I need it for Theorem 3.

At the moment, I'm fairly convinced what I mentioned is only a subspace of $LaTeX Code: B(\\mathbb{R})$ ...

There's more work to be done on just what a group homomorphism from $LaTeX Code: \\mathbb{Q}$ to $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ looks like. [:(]

**Hurkyl** · Feb19-04, 06:27 PM

All right, I got it this time.

The mistake I made was assuming that if $LaTeX Code: f : \\mathbb{Q} \\rightarrow \\mathbb{R} / \\mathbb{Z}$ is a homomorphism, then $LaTeX Code: f(r) \\rightarrow 0$ as $LaTeX Code: r \\rightarrow 0$ .

However, we can still catalog these homomorphisms!

Define $LaTeX Code: [f]_0 := f(1) \\quad (0 \\leq [f]_0 < 1)$

Now, consider the possible values of

. Because we know the value of

, there are only two possibilities for

. We define the next statistic implicitly by

LaTeX Code: f(1/2) = (1/2) (f(1) + [f]_1)

.

Then, given

, there are three possibilities for

. We define the next statistic by

LaTeX Code: f(1/6) = (1/3) (f(1/2) + [f]_2)

.

In general, $LaTeX Code: f(1/n!) = (1/n) (f(1/(n-1)!) + [f]_{n-1})$ .

Allow me to emphasize that $LaTeX Code: [f]_0 \\in \\mathbb{R}$ , but $LaTeX Code: [f]_i \\in \\mathbb{N}_i$ for

.

Given these statistics

, we can recover the function through the formula:

$LaTeX Code: f(\\frac{p}{q!}) = \\frac{p}{q!} \\sum_{i=0}^{q-1} (i! [f]_i) $

Notice that higher order terms (

) of the sum are irrelevant, because we're working mod 1.

With a little effort, we can see that each sequence of statistics describes a unique homomorphism, and each homomorphism has a sequence of statistics, so we have a catalog of all homomorphisms from $LaTeX Code: \\mathbb{Q}$ to $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ .

So what is the topology of this space? Well, as per my previous post, convergence in this space is given by pointwise convergence in $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ , so what does this mean?

If $LaTeX Code: f_i \\rightarrow f$ , then for any $LaTeX Code: q \\in \\mathbb{Q}$ , we have that $LaTeX Code: f_i(q) \\rightarrow f(q)$ .

Plugging in the formula, this means (after a little rearranging, and applying the homomorphism property) for all positive integers q, the sum

$LaTeX Code: \\frac{1}{q!} \\sum_{i=0}^{q-1} ([f_n]_i - [f]_i)$

must converge to 0 as n increases.

This usually means that $LaTeX Code: [f_n]_0 \\rightarrow [f]_0$ as $LaTeX Code: n \\rightarrow \\infty$ , and that each

is eventually equal to

, for

. However, things get messy when

...

To see what is going on, allow me to comment on this representation. There is something called (IIRC) the "factorial base" representation of nonnegative integers. Recall that the decimal representation of an integer is given by:

$LaTeX Code: n = \\sum_i 10^i n_i $

If you're careful, on ranges, one can use other things instead of

for scaling; the factorial base uses

as the scale factor. With a little playing around, you can see that given the right bounds on the terms, each positive integer can be written uniquely as

$LaTeX Code: n = \\sum_i i! n_i $

You can see how this relates closely to the formula given above. If we add a fractional term to the factorial base representation, then we can uniquely write any nonnegative real number!

For example, we can write the integer 37 as: 1201
And you can check that 37 = 1 * 1! + 0 * 2! + 2 * 3! + 1 * 4!

Notice that the ordinary addition algorithm works in this representation. For example, 1201 + 1 = 1210 == 38. (Note that because you exceeded the limit of 1 in the 1!'s place, you have to carry to the 2!'s place)

However, the sequence of interest doesn't have to be eventually zero, so this is somewhat of a transfinite generalization of the factorial base representation. The important thing to note, though, is that how incrementing and decrementing work just like in ordinary arithmetic; you have to be sure to borrow and carry when appropriate.

Now, back to the messy situation. When

, we lie on a boundary. If we move slightly forward, all of the other

remain unchanged, but if we move slightly backward, we have to "borrow". For example:

1201 + .1, 1201 + .01, 1201 + .001

aproaches 1201 + 0

But going the other way:

1200 + .9, 1200 + .99, 1200 + .999

also approaches 1201 + 0.

Now, there's still more of a mess! Consider this sequence:

11201 + 0, 101201 + 0, 1001201 + 0, 10001201 + 0, ...

Note that this also converges to 1201 + 0!!!

(and I haven't even tried to write any of the messes we can have with a string of digits that is not eventually 0)

Anyways, the point is that we now have a catalog of all homomorphisms $LaTeX Code: f\\phi: \\mathbb{Q} \\rightarrow \\mathbb{R} / \\mathbb{Z}$ . The spirit of my treatment carries on unchanged. We want to consider $LaTeX Code: \\mathbb{R}$ as a vector space over $LaTeX Code: \\mathbb{Q}$ , and use the decomposition into $LaTeX Code: \\mathbb{Q}$ -linear combinations of basis elements, and then we apply one of the ugly things we discovered in this post to each of the dimensions of $LaTeX Code: \\mathbb{R}$ , and in this way we generate all homomorphisms from $LaTeX Code: \\mathbb{R}$ to $LaTeX Code: \\mathbb{R} / \\mathbb{Z}$ .

Yuck!

**Hurkyl** · Feb21-04, 11:18 PM

Ok, let me formally present this representation of $LaTeX Code: \\mathrm{Hom}(\\mathbb{Q}, \\mathbb{R}/\\mathbb{Z})$ , complete with the appropriate topology.

Let us define the set

to be the set of all sequences $LaTeX Code: \\{b_i\\}_0^{\\infty}$ where $LaTeX Code: b_0 \\in \\mathbb{R}$ , $LaTeX Code: 0 \\leq b_0 < 1$ , and for

, $LaTeX Code: b_i \\in \\mathbb{N}$ , $LaTeX Code: 0 \\leq b_i \\leq i$

We will write elements of

in the following way:

$LaTeX Code: \\ldots ,b_3,b_2,b_1.b_0$

where the commas are optional symbols used for clarity. So, for instance, the sequence

LaTeX Code: b_0 = 1/2, b_1 = 1, b_2 = 0, b_3 = 1, b_i = 0 (i > 3)

will be written as

$LaTeX Code: \\ldots00101.5$

Or, in the case where the sequence is eventually zero, we use the shorthand

We can define addition and subtraction on these things by analogy with the way we define addition and subtraction on the usual decimal (or n-ary) representation of real numbers. So, for example:

$LaTeX Code: \\begin{array}{r} \\ldots 54310.5 \\\\ + 1.5 \\\\ \\hline \\ldots 54321.0 \\\\ \\end{array} $

Note that we had to carry twice; first because

in the 0th place, and second because

in the 1st place.

The set

with this structure for addition forms an additive group.

There is a natural embedding of $LaTeX Code: \\mathbb{R}$ in

, which I will give via the reverse map:

If $LaTeX Code: \\{ b_i\\} \\in M$ is an eventually zero sequence, then we can map it to a real number by

$LaTeX Code: \\{ b_i \\} \\rightarrow \\sum_{i=0}^{\\infty} b_i i! $

This gives us any nonnegative real number. We can get the negative real numbers via taking additive inverses of eventually zero sequences. (alternatively, you could use the analog of two's complement notation to give them explicitly)

So, for instance:

$LaTeX Code: 101.5_M = 0.5 * 0! + 1 * 1!+ 0 * 2! + 1 * 3! = 7.5_{\\mathbb{R}} $

$LaTeX Code: \\ldots,4321.0_M = -1_{\\mathbb{R}} $

(check by adding 1 to $LaTeX Code: \\ldots,4321.0$ )

We also want to consider quotient spaces of

by truncating it after a finite number of digits. Note that if we truncate elements of

to n digits long, then

is isomorphic to $LaTeX Code: \\mathbb{R} (\\mathrm{mod} n!)$ (via the truncated form of the mapping given above)

Now elements of

are supposed to represent homomorphisms from $LaTeX Code: \\mathbb{Q}$ to $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ . They do this by "multiplication"; if we take a rational number written in the form

and $LaTeX Code: \\{ b_i \\} \\in M$ , we can use the above mapping to formally multiply:

$LaTeX Code: \\{ b_i \\} \\left( \\frac{p}{q!} \\right) = \\left( \\sum_{i=0}^{\\infty} b_i i!\\right) \\frac{p}{q!} $

Note that because the range is $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ , the terms with $LaTeX Code: i \\geq q$ don't contribute anything to this sum. Thus, we can rewrite as

$LaTeX Code: \\{ b_i \\} \\left( \\frac{p}{q!} \\right) = \\left( \\sum_{i=0}^{q-1} b_i i!\\right) \\frac{p}{q!} $

and this works for any element of

, not just eventually zero ones.

(Notice that the fact the domain of the homomorphism is $LaTeX Code: \\mathbb{Q}$ is essential!)

(Also note that if we add elements in

then convert the result into a homomorphism, the result is the same as if we converted the elements into homomorphisms first then added them)

Finally, I can describe the topology of

. Recall that we desire convergence of a sequence in $LaTeX Code: \\mathrm{Hom}(\\mathbb{Q}, \\mathbb{R}/\\mathbb{Z})$ to be pointwise convergence.

This is the same as saying that a sequence converges in

iff it converges in every truncation of

.

And finally, I can describe what neighborhoods look like:

To construct a neighborhood of $LaTeX Code: m \\in M$ we first construct the set $LaTeX Code: S_{\\epsilon} = \\{ m + x | -\\epsilon < x < \\epsilon\\}$ for some real number $LaTeX Code: \\epsilon > 0$ . Then, we construct the set $LaTeX Code: U_{N, \\epsilon}$ which consists of all elements of

that agree with elements of $LaTeX Code: S_{\\epsilon}$ on the first

digits.

To demonstrate the two "orthogonal" directions of convergence, both of these sequences converge to 0:

2.5, 1.1, 0.01, 0.001, ...
1.0, 10.0, 100.0, 1000.0, ...

One can imagine

being constructed by taking infinitely many copies of the real line and attaching them end to end wrapping them around a torus. Sequences have to converge both by their position in the real line and have to converge to the right copy of the real line!

Now, I can say explicitly what the Bohr compactification of the real line ( $LaTeX Code: B(\\mathbb{R})$ ) looks like.

Choose a basis

of the $LaTeX Code: \\mathbb{Q}$ -vector space of the real numbers. Elements of $LaTeX Code: B(\\mathbb{R})$ are functions from

to

.

If $LaTeX Code: \\sum_{b \\in B} q_b b = r$ is an element of $LaTeX Code: \\mathbb{R}$ (the

are rational, and only a finite number of them are nonzero), and we're given $LaTeX Code: f \\in B(\\mathbb{R})$ , then we compute

by:

$LaTeX Code: f(r) = \\sum_{b \\in B} f(b)(q_b) $

(Notice this is well-defined since the summand can only be nonzero a finite number of times)

(So basically, for each dimension of $LaTeX Code: \\mathbb{R}$ we choose a way in

for that dimension to transform, and we combine dimensions in an additive way)

Convergence in $LaTeX Code: B(\\mathbb{R})$ is given by pointwise convergence. Or, alternatively, you can say $LaTeX Code: B(\\mathbb{R})$ has the topology it inherits from being the Cartesian product of uncountably many copies of

.

Finally, there is a natural embedding of $LaTeX Code: \\mathbb{R}$ in $LaTeX Code: B(\\mathbb{R})$ given by $LaTeX Code: x \\rightarrow R_x$ where $LaTeX Code: R_x : B \\rightarrow M : b \\rightarrow bx$ . (The "multiply by x" map)

So there you have it, this is what the Bohr compactification of the real numbers looks like. It's an awful terrible thing, and I think the only usefulness of this exercise is to appreciate how horrible it really is. [:)] I imagine for any practical purpose you can just say that elements of $LaTeX Code: B(\\mathbb{R})$ are additive homomorphisms from $LaTeX Code: \\mathbb{R}$ to $LaTeX Code: \\mathbb{R}/\\mathbb{Z}$ and be done with it!

edit: Remembered to mention how the negative real numbers fit into

removed a description of neighborhoods of

that was missing a detail.

**marcus** · Feb21-04, 11:47 PM

I'm actually a little awed
thanks
nice to have a definite construction
instead of just the abstract version