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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This entry is obsolete, having been superseded by a new entry.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) ]
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See Also: group, topological ring, Birkhoff-Kakutani theorem, categories of Polish groups and Polish spaces, topics in algebraic topology
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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 6175 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.
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Discussion
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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
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