PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (46)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
topological group (obsolete) (Definition)

This entry is obsolete, having been superseded by a new entry.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.



"topological group (obsolete)" is owned by rspuzio. [ full author list (2) | owner history (2) ]
(view preamble | get metadata)

View style:

See Also: group, topological ring, Birkhoff-Kakutani theorem, categories of Polish groups and Polish spaces, topics in algebraic topology
Log in to rate this entry.
(view current ratings)

Cross-references: Hausdorff, map, product topology, continuous, group operation, topology, group, topological group
There are 3 references to this entry.

This is version 6 of topological group (obsolete), born on 2002-01-22, modified 2006-03-19.
Object id is 1543, canonical name is TopologicalGroup.
Accessed 6180 times total.

Classification:
AMS MSC: 22A05 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Structure of general topological groups)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy
can't accept object transfer by rspuzio on 2006-03-22 11:37:14
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.
[ reply | up ]
against pretentious formal notation by stevecheng on 2006-03-18 14:45:58
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
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)