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Hometopological group (obsolete)

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# topological group (obsolete)

This entry is obsolete, having been superseded by a new entry. It is being retained for a short while because of the attached thread.

A *topological group* is a triple $(G,\cdot,\mathcal{T})$ where $(G,\cdot)$ is a group and $\mathcal{T}$ is a topology on $G$ such that under $\mathcal{T}$, the group operation $(x,y)\mapsto x\cdot y$ is continuous with respect to the product topology on $G\times G$ and the inverse map $x\mapsto x^{{-1}}$ is continuous on $G$.

Many authors require that the topology be Hausdorff.

Related:

Group, TopologicalRing, BirkhoffKakutaniTheorem, CategoryOfPolishGroups, AlgebraicTopology

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

22A05*no label found*

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## Comments

## against pretentious formal notation

I'm not submitting this as a correction to the entry

http://planetmath.org/encyclopedia/TopologicalGroup.html

because the opinion below, I expect, will be controversial,

and the above entry is not even a particularly egregious example

of what I'm going to criticize.

But I have always wanted to bring the topic up for discussion.

======================================

The use of pretentious formal notation

======================================

Why is it necessary to write

``A topological group is a triple $(G,\cdot,\mathcal{T})$ ...''

instead of

``A topological group is a group equipped with a topology

under which the group multiplication and the group inverse are continuous mappings.''

or if you think that is not precise enough:

``A topological group is a group, equipped with a topology $\mathcal{T}$ under which the group

multiplication $(x,y) \mapsto x \cdot y$

is continuous in the product topology $\mathcal{T} \times \mathcal{T}$,

and the inverse $x \mapsto x^{-1}$ is continuous

in the topology $\mathcal{T}$.''

My point is, why is it necessary to introduce the notion

of triples? Is the syntax, that you put $G$ and $\cdot$ before $\mathcal{T}$, really relevant to the concept of topological groups?

(a tuple is order-sensitive)

Are you using extra mathematical symbols to make yourself look

sophisticated, like this entry?

http://planetmath.org/?op=getobj&from=objects&id=7374

which seems to have been written by mathematician-wannabe

computer scientists

Let me emphasize how ridiculuous this is.

This is how computer-science-formalese would define the English word ``library'':

``A library is a 2-element tuple (library, books),

where library is an object representing the building

holding the books and books is a set of objects representing books. ''

Of course, it would be ridiculuous to suggest that we should always prefer informal notation to formal notation. But in this case,

I think a good rule of thumb would be:

``Do you need to manipulate the objects _as tuples_?''

For example, we can reasonably (and frequently we do)

talk about equality of Cartesian coordinates:

``two point on the plane $(x_1,y_1)$ and $(x_2,y_2)$ coincide

if and if $(x_1,y_1) = (x_2, y_2)$

(meaning that $x_1 = x_2$ and $y_1 = y_2$)''

On the other hand, do you ever need to use an equation like the following

$(G, \cdot, \mathcal{T}) = (H, \cdot', \mathcal{S})$

If not, then I'd suggest that we chuck the tuple notation.

It's irrelevant.

// Steve

## Re: against pretentious formal notation

Addendum:

Another good rule of thumb to when to use a tuple or not:

Do you need to project the object, or

index the tuple with a numerical index?

For example:

$\pi_k(x)$ or $x_k$ extracts the $k$th coordinate of the tuple $x$

One may also ask if we need to work on the tuple as a whole

e.g. $x \cdot y$ takes the dot product of the row vectors $x$ and $y$

$\lVert x \rVert$ takes the norm of $x$

and so forth. If so, then obviously, use the tuple.

The use of the tuples in formalese fails these tests.

Would you write: topological group $H = (G, \cdot, \mathcal{T})$,

$\mathcal{T} = H_3$ !?

## Re: against pretentious formal notation

I'm a fan of having both formalizations and informalizations.

They're both useful.

apk

## Re: against pretentious formal notation

I think you should read the "spirit of ubuntu" posting I just wrote up

(http://planetmath.org/?op=getmsg&id=10087). Why are you using the

word "pretentious" in this discussion? A triple is not pretentious.

Perhaps some justification for the notation should be given (e.g. why,

precisely, is G listed first in the triple, why are triples used in

the first place).

In this case, I would simply ask the author to explain this choice of

notation, rather than asserting that this choice is pretentious and

making an unfounded allegation ("Are you using extra mathematical

symbols to make yourself look sophisticated, like this entry?").

You can approach the matter more respectfully by posing the essentially

technical question: why are you using triples?

## Re: against pretentious formal notation

the tuple notation is quite standard in mathematics. i find

it quite easy to take in at a glance what is being defined.

i don't think an author needs to explain why he uses

common notation. instead, perhaps there should be a discussion in

the mathematical community of why this notation is used.

personally, i think a good entry has both a formal and

informal definition, as aaron said.

-kyle

## Re: against pretentious formal notation

>

> Why is it necessary to write

>

> ``A topological group is a triple $(G,\cdot,\mathcal{T})$

> ...''

>

I think this should be read as follows: A topological group

is a new mathematical object depending on three objects;

a group G with product $\cdot$, and a topology $T$ on $G$ such

that ...

In other words, the triple "notation" is just a way

of emphasizing what the new defined object depends on. I

don't like it very much, although I have probably used it myself

sometimes. For example, of:

(X,d) metric space.

Let $(X,d)$ be a metric space.

Let $X$ be a metric space with metric $d$.

the first one is convenient on a blackboard, but

latter is probably the best writing.

> ``A topological group is a group equipped with a topology

> under which the group multiplication and the group inverse

> are continuous mappings.''

>

To this I would add a note about the product topology.

Multiplication should be continuous as a map $G\times G\to G$

under the product topology for $G\times G$. On the other

hand, if the definition would be followed by some examples,

and properties this could be mentioned there.

> Of course, it would be ridiculuous to suggest that we should

> always prefer informal notation to formal notation. But in

> this case,

> I think a good rule of thumb would be:

>

> ``Do you need to manipulate the objects _as tuples_?''

>

This is a good point. A common mistake is to introduce

a lot of notation (intended for the proof) in the theorem

formulation. THis just clutters up the formulation

and makes it difficult to grasp.

## Re: against pretentious formal notation

stevecheng writes:

> I'm not submitting this as a correction to the entry

> http://planetmath.org/encyclopedia/TopologicalGroup.html

> because the opinion below, I expect, will be controversial,

> and the above entry is not even a particularly egregious

> example of what I'm going to criticize.

So why didn't you post it in a forum instead of attaching it to this entry? If you were hoping that the author would see it here, you should note that he hasn't even logged in for over a year. If you really want him to know what you think of his triples, you could try e-mailing him.

You really couldn't have picked a worse entry to attach your post to: I adopted this entry yesterday, and I intend to delete it, replacing it with a new one. I am doing this (rather than just modifying the existing entry) in order to make it clear that I am the sole author (since there is nothing in the previous author's version that I want to keep). This way the entry can easily be relicensed when PlanetMath moves away from the flawed FDL, as I hope it will.

I've already written the new entry:

http://planetmath.org/encyclopedia/TopologicalGroup2.html

When I delete the old entry this whole thread will presumably disappear. I will leave it a few hours before doing this. Maybe Aaron can save the thread by moving it to a suitable forum.

## more refined argument against ``is a triple'' -type definiti...

Alright, I admit I wrote that piece on impulse, after seeing

yet another ``is a triple...'' definition and being

annoyed at it. I really should have refined my opinion

--- although most replies have not actually addressed the points

of my argument other than essentially ``I agree'', ``I disagree''.

I apologize for the offence that I caused.

Now I want to state my refined opinion:

After thinking about it a bit, I'm don't have anything against

using a notation of the form $(X, \Sigma, \mu)$ (to take

a common example) to inform the reader of what the letters mean.

For example,

``Let $(X, \Sigma, \mu)$ be a measure space. Then ...''

What I have a problem with is with definitions like:

``A measure space _is a triple_ $(X, \mu, \Sigma)$ where''

for the phrase _*is* a triple_ seems to emphasize the syntax

rather then the concept being defined,

and all the important concept is relegated to a sentence

subclause. The canned phrase _is a tuple_ is essentially

a cop-out when the author has nothing meaningful to say

about the tuple $(X, \mu, \Sigma)$ itself,

but only wants to talk about its constituent parts $X$,

$\Sigma$, and $\mu$, and that the parts are to be used together.

I don't think in the majority of cases this is

what the author wants; he only writes that because he is often

exposed to this kind of writing and think of no reason not to imitate it, and/or he is not thinking carefully about what he is writing.

My accusation that it was pretentious was really a rhetorical question, although I regret I directed it too specifically to the Toplogical groups entry that I made an example of.

It is a rhetorical question that I sometimes ask myself when I proofread my writing --- is it too pretentious written?

That question is also a jab at those instances

(not the Topological groups entry)

where the excessive formality seems to be deliberate

rather than the author not realizing that the writing can be improved.

As someone has pointed out, the triple _notation_

is fairly standard. I agree. But is the

triple-based _definition_ standard?

I haven't looked so thoroughly to be sure one way or another.

In any case, my opinion is that it is a bad style of writing.

It is not wrong _mathematically_, of course.

On the other hand, there are lots of mathematics books out there,

written by experts whose rigour and reputation I don't haven't the foggiest doubt of, but whose style of exposition is terrible, and difficult

for a reader who is not already familiar with the work to follow.

Even something as basic as typesetting the work

with margins separating section and theorem numbers from the main text, and spacing between different sections of the text,

--- all would make the text easier to read ---

is not done, even though this is the responsibility of the publisher side rather than the author.

I think it is important, for the criticism, to make the distinction between the triples *notation* and *definition*,

and this is not a mere concession.

Notation is a means of communicating, as concise and precise

as possible. It should also be as unified as possible

(to reasonable limits). The $(X,\Sigma,\mu)$ notation

serves this purpose. But it could have well have been written

$\lbrace X, \Sigma, \mu \rbrace$, which would mean essentially the same thing, although it is not usually written this way.

And as such, it is important not to codify a choice

of notation in the definition using phrases like ``is a triple...''.

As another example,

the derivative (function) of $f$ is not the symbol the $f'$, nor

the symbol $\frac{df}{dx}$, it is a function, denoted by $f'$, such that

$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$.

In the case of the measure space example, I would consider writing:

``A measure space $(X, \Sigma, \mu)$ is a set X,

together with a sigma-algebra $\Sigma$ on $X$, and a measure $\mu\colon \Sigma \to \mathbb{R}$.''

Note that I say that a measure space *is* a set X.

This corresponds with the English usage, and informal usage;

after all, the word ``space'' is in the term ``measure space'',

and it is the noun that is being modified by the term ``measure''.

It is incongruent if you think the word *is* means *equal*,

(though compare that with the statement ``$G$ is a group'').

Another wording that avoids the word *is* is the following:

``A measure space $(X, \Sigma, \mu)$ consists of a set X,

a sigma-algebra $\Sigma$ on it, and a measure $\mu$

on that sigma-algebra.''

Let me make a final analogy: that of the $\forall$ notation.

Everybody knows about it; there isn't even any misunderstanding

if an author uses it in a mathematical exposition (not about formal logic). But is it a good idea to use it as a substitute

for the English phrase ``for all'' everywhere?

Most people would say no. It adds no meaning, not all

that much more concise. It's just not good stylistically.

To summarize, my objection to the ``is a triple''-type definition

falls along the same lines: it is not good style.

// Steve

## Re: more refined argument against ``is a triple'' -type defi...

> After thinking about it a bit, I'm don't have anything

> against

> using a notation of the form $(X, \Sigma, \mu)$ (to take

> a common example) to inform the reader of what the letters

> mean.

> For example,

>

> ``Let $(X, \Sigma, \mu)$ be a measure space. Then ...''

>

Agreed. But, on PM, a "measure space" is denoted by

$ (E, \mathcal{B}, \mu)$.

See

http://planetmath.org/encyclopedia/CountablyAdditive2.html

So, on PM one needs to be careful what notation one uses.

>

> ``A measure space _is a triple_ $(X, \mu, \Sigma)$ where''

>

..

..

> ``A measure space $(X, \Sigma, \mu)$ is a set X,

> together with a sigma-algebra $\Sigma$ on $X$, and a measure

> $\mu\colon \Sigma \to \mathbb{R}$.''

>

> Note that I say that a measure space *is* a set X.

I would agree that the first is better.

> Let me make a final analogy: that of the $\forall$ notation.

> Everybody knows about it; there isn't even any

> misunderstanding

> if an author uses it in a mathematical exposition (not about

> formal logic). But is it a good idea to use it as a

> substitute

> for the English phrase ``for all'' everywhere?

> Most people would say no. It adds no meaning, not all

> that much more concise. It's just not good stylistically.

Many times you can just delete it. For example, compare

$$

x\in \mathcal{F}(y), \quad \forall y\in C.

$$

and

$$

x\in \mathcal{F}(y), \quad y\in C.

$$

>

> To summarize, my objection to the ``is a triple''-type

> definition

> falls along the same lines: it is not good style.

Rudin (Real and complex analysis) also discusses the

"triple"-notation. As an example he describes the

real numbers as the quadruple $(R,\cdot,+,<)$.

## Re: more refined argument against ``is a triple'' -type defi...

> Rudin (Real and complex analysis) also discusses the

> "triple"-notation. As an example he describes the

> real numbers as the quadruple $(\mathbb{R},â‹…,+,<)$.

Good that you mention this example.

I would argue that it is formal dressing that doesn't add anything

to a more descriptive, syntax-independent, definition.

First, the notation is actually not so precise as

it might first seem.

After all, it doesn't say anything about the symbols

$1, 0, /, \cdot^{-1}, >, \neq$, etc.

even though, say $\cdot^{-1}$ could conceivably

be taken as meaning subtraction. And would you remember

that $\cdot$ goes before $+$, and what would you think

of this statement:

``The real numbers are the quadruple $(\mathbb{R}, +, \cdot, <)$.

I have switched the symbols $+$ and $\cdot$! Would that be a mere typo, or does it indicate something significant?

(i.e. is the ordering of those symbols

and that they have been put in a tuple, really

relevant to the definition of real numbers?)

And you couldn't quite say ``5 belongs to the real numbers'', can you?

Because real numbers aren't the set $\mathbb{R}$ according to this definition; rather it is this funny tuple.

// Steve

## Re: more refined argument against ``is a triple'' -type defi...

> Another wording that avoids the word *is* is the following:

>``A measure space $(X, \Sigma, \mu)$ consists of a set X,

>a sigma-algebra $\Sigma$ on it, and a measure $\mu$

>on that sigma-algebra.''

I'm surprised you find this palatable. Doesn't this implicitly define a measure space to be a set consisting of three elements? I see little implicit difference between this definition and the following:

``A measure space is a set ${X, \Sigma, \mu}$ consisting of a set X,

a sigma-algebra $\Sigma$ on it, and a measure $\mu$

on that sigma-algebra.''

The one (admittedly non-trivial) distinction is that in the first case there is an implicit ordering in the notation, whereas in the latter, as long as you know which object is which, you could write it in any order. If that's your main point, then I would certainly concede that an unfortunate byproduct of the tuple notation is that it implies an ordering of the required data. On the other hand, mathematical writing is filled with ad hoc conventions for organizing data (consider left- and right-actions, and composition of permutations), none of which are typically regarded as bad style.

In general, I strongly prefer the tuple definition. I think it's important to remember exactly what pieces of data you get when you specify an object, and then to think of an object just as that collection of data, at least until the concept is internalized. One has no hope, for example, of understanding a scheme, if one simply thinks of it as a set or a topological space (since then the spec of every field is the same scheme!). I think thinking of a scheme (and many other objects as well) as a collection of information organized in a tuple is a good step toward an actual understanding of the concept.

Cam

## Re: more refined argument against ``is a triple'' -type defi...

I'm glad you brought up the following point, giving me another opportunity to sharpen my argument.

> >``A measure space $(X, \Sigma, \mu)$ consists of a set X,

> >a sigma-algebra $\Sigma$ on it, and a measure $\mu$

> >on that sigma-algebra.''

>

> I'm surprised you find this palatable. Doesn't this

> implicitly define a measure space to be a set consisting of

> three elements?

The difference is in style. (I explain that below.)

> I see little implicit difference between

> this definition and the following:

>

> ``A measure space is a set ${X, \Sigma, \mu}$ consisting of

> a set X,

> a sigma-algebra $\Sigma$ on it, and a measure $\mu$

> on that sigma-algebra.''

The difference is that the first quote de-emphasizes

the set or tuple used to hold the three objects under discussion,

because it is mere syntax. When you read the quote, the verb ``consists'' alerts you to the fact that some objects will follow,

and those objects are important because they are

(grammatical) objects. Whereas

``blah is a tuple, where (....)''

says that blah *is* a tuple, and all the important information

is buried inside a subsidiary "where" subclause.

Surely you would agree that

there is a difference between a container,

and the objects held inside that container. We want

to focus the attention on the objects held on the container

(in this case, the set $X$, the sigma-algebra $\Sigma$,

and the measure $\mu$), not on the set

$\lbrace X, \Sigma, \mu\rbrace$

or the tuple $(X, \Sigma, \mu)$,

which in most situations is just syntax.

Let me give another example. Consider the definition:

``A real number is an equivalence class of Cauchy sequences of rational numbers, where (...definition of the equivalence...)''

This is alright as a definition if it is to be used in an exposition

that constructs the real numbers out of Cauchy sequences of rational numbers. But would this be a good definition for a general encyclopedia entry? I would think not.

The fact that when you ``look inside'' a real number

you see Cauchy sequences of rational numbers is irrelevant

to what the real numbers *are*. I could have used some other construction where real numbers are decimal sequences. The problem

with the ``*is* a tuple'' definitions is similar: the definition focuses undue attention on the syntax.

// Steve

## Re: more refined argument against ``is a triple'' -type defi...

> The fact that when you ``look inside'' a real number you see Cauchy

> sequences of rational numbers is irrelevant to what the real numbers

> *are*.

Alright then, what is a real number, really?

The only way I know to answer this question is to first pick some

axiomatic framework and work within it. In one framework, real

numbers are equivalence classes of Cauchy sequences. In another

framework, they are Dedekind cuts. In a third, they are objects which

satisfy certain properties.

Of course, one should complememnt this with a description of how to

relate these different axiomatic descriptions. Which sequence

corresponds to which set of rational numbers? How does one assign a

Cauchy sequence or a Dedekind cut to an object satisfying the axioms?

In a way, this is much like motion in Physics. To describe the motion

of an object, we need to first pick a frame of reference, then say

what is going on in that reference frame and be careful to not

indiscrimainately mix up things from different reference frames

without first transforming them into a common reference frame.

Just as it is meaningless to ask whether an object is at rest except

witin the context of a particular reference frame, so I would say that

this business of looking inside real numbers is similarly meaningless

except in the context of a specific axiomatic framework. In the first

system, you see sequences; in the second you sets of rational numbers;

in the third, you don't see anything becasue looking inside is

undefined.

## Re: more refined argument against ``is a triple'' -type defi...

I would say that some of the isssues raised here aren't matters of

style, they're matters of logical precision and rigor.

> On the other hand, mathematical writing is filled with ad hoc

> conventions for organizing data (consider left- and right-actions,

> and composition of permutations), none of which are typically

> regarded as bad style.

While these choices are arbitrary (because everything is covariant

under permutation of terms in the tuplet) it is necessarry to make

some arbitrary choice of convention and stick with it consistently.

For instance, a ring is defined by a triplet (R, +, x), where "+" and

"x" are binary operations on the set R. While it doesn't matter

whether I choose to write "(R, +, x)" or "(R, x, +)", I need to make a

choice and stick with it consistently. If I switch "+" with "x" in

the disstrtibutive law for instance, I get a false statement.

> In general, I strongly prefer the tuple definition. I think it's

> important to remember exactly what pieces of data you get when you

> specify an object, and then to think of an object just as that

> collection of data, at least until the concept is internalized.

More importantly, this is a matter of proper axiomatic presentation.

In order to describe an axiomatic system, one needs to know the

primitive terms and the axioms. The items in these triplets are the

primitive terms. Unless one specifies the primitive terms, one has

not completely specified one's system.

I think that this entry is well-written and rigorous and do not

understand why it should be deleted. As I see it, there is no reason

for not having both entries --- if someone wants a more intuitive

definition with more words than symbols, one can look at one entry,

while if one wants a precise, formal definition of the same, one has

the other entry. Both sentries serve useful purposes and I hope Yark

will relent and let both entries stay. If there is an issue of two

entries, then I would say that the proper solution is to add some sort

of metadata which would let people filter based on their preferences.

In addition to such distinctions as formal vs. expository, one could

also hav such things as level of difficulty (beginner, intermediate,

advanced), etc.

## Re: more refined argument against ``is a triple'' -type defi...

rspuzio writes:

> Both sentries serve useful purposes and I hope

> Yark will relent and let both entries stay.

What do you think is useful about the old entry? If I include the triple notation in the new entry, would that make the old one redundant in your eyes?

In general I think that duplicate entries on PlanetMath are a good thing (though obviously it's better to write an entry about something new rather than something that's already covered). But I had no intention of duplicating this entry, and I doubt anyone would even have noticed that I had replaced the old entry with a new one if I hadn't had to keep the old one around for while to prevent this thread from disappearing. (Is it really impossible for an admin to move the thread to a forum?)

I am not going to keep this entry, but I could transfer it to you if you really think it's worth keeping.

## Re: more refined argument against ``is a triple'' -type defi...

> I am not going to keep this entry, but I could transfer it to you if

> you really think it's worth keeping.

I'll take it.

## can't accept object transfer

Yet another problem surrounding this benighted entry --- Yark offered

me ownership, of the object, I clicked "accept", the computer said I

was given ownership, but did not actually transfer ownership to me.

This is the second time I havetried to accept the transfer of

ownership --- the first time was a few adays ago; I thought all went

wel but got another e-mail posting askiing me to accept ownership of

the object.

The reason I am posting this here is because of the bug Yark noticed

with the bug reporting system.

## Re: can't accept object transfer

I assumed something had gone wrong with the first transfer request, as I hadn't got any notice back about acceptance or rejection of the transfer, and the entry was still owned by me. So I sent a second request today, which is why you got two. Both transfer requests appeared to go normally from this end, but I didn't receive either of the acceptance notices that I should have got.

## Re: against pretentious formal notation

Even though this discussion is months old, let me add my input to it, because quite a few of my own entries use the "pretentious" triple notation, even though this entry in particular is not mine.

It is completely clear from the discussion that you (stevecheng, the originator of this complaint) have never studied anything that I would consider to be extremely advanced mathematics. In saying this, I do not mean to denigrate your knowledge or to discourage you in any way. Indeed, I strongly encourage you to continue learning more mathematics, because it is the best way for you to outgrow your aversion to triples definitions. I guarantee you that, as you advance, there will come a point in your studies when you will find yourself *unable* to learn new definitions unless you use triples.

The reason is that, as definitions increase in complexity, it becomes impossible for the human mind to categorize all the elements of the definition properly unless the individual components of the definition are fully laid out, in a precise fashion, using "pretentious" formal notation. If you are still at the level of topological groups or measure spaces, then I can see how you might think triples are unnecessary. However, once you get to definitions such as group schemes or algebraic stacks, most mathematicians (with _rare_ exceptions) need to have a definition that specifies every component of the object, with formal notation, in order to even have any hope of learning what the object means.

If it should turn out that you are one of the exceedingly rare individuals who can learn group schemes or algebraic stacks without using tuple notation, then, well, people like that are such a small minority that it would be counterproductive to target PlanetMath towards those people at the expense of the majority.

The reason I use tuple notation even in comparatively simple definitions such as measure space is because I believe it is good practice to get readers into the habit of accepting formal notation in definitions, in order to prepare them for later on in their studies when such techniques become indispensible.

## Re: against pretentious formal notation

It might be said that pretentious, formal mathematics requires pretentious formal notation!

But joking aside, I too believe that the initial seeming artificiality of much of mathematical notation is greatly compensated by the benefits of it.

However, it is certainly possible to overdo it. I personally do detest a mode of mathematical writing that insists on using strings of symbols where a few phrases would do much better. They might be even more evocative of the spirit underlying the concept(s) being presented.

Overly formalistic writing reminds me of the statement made by philosopher / epigrammatist La Rochefoucauld (?) that "language was given to men that they might conceal their thoughts"! (I'm almost certainly not quoting very faithfully).