# topological embedding

## Primary tabs

Synonym:
imbedding
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### bi-directional updating

I see that there are a whole bunch of entries showing up nowadays that are based on Wikipedia entries. This is fine, but it suggests that it might be time to reconsider the business of bi-directional updating. Rather than copying entries one-by-one on an ad hoc basis, it might be less work in the long run to work out a scheme which will do this automatically, or at least partially utomatically.

### Re: bi-directional updating

I perfectly agree. Many entries are more complete in wikipedia than in planetmath. But I don't think that there is one-to-one correspondence between wiki entries and PM entries and that they could be simply copied. To my knowledge, Wiki entry usually contains also generalizations of the defined notion and also the short introduction intended for "non-matematicians".

As for this entry - it was intended as a "seed" - I left open access for editing. I didn't like that in topology-related entries the usage of the word "embedding" was automatically linked either to injective function or to embedding of manifolds. And since there was the entry in wiki, I used it.

### Re: bi-directional updating

Copying or copy-and-modifying from Wikipedia is NOT fine in my view, since until
the site admins (for both sites, actually) take action, such copying is not going to be GFDL
compliant. However, the current article is so loosely "based"
on the corresponding Wikipedia entry, I am surprised that rspuzio
was able to notice, correctly, that the OP based the article
on the WP source (the fact that there is only one reference notwithstanding).
in the technical, legal, sense of the word.

I also agree that it would be good to work out a technologically
(and legally) sound scheme for semi-automatic bidirectional
updating. Of course, this idea has been under discussion for a
while & indeed, related work is taking place more or less as
we speak.

### Re: bi-directional updating

Just to point you that several admins on wikipedia are involved on a PM-WP exchange, look at http://en.wikipedia.org/wiki/WP%3APMEX
May I ask you why do you think it would not be GFDL compliant?
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: bi-directional updating

> However, the current article is so loosely "based"
> on the corresponding Wikipedia entry, I am surprised that rspuzio
> was able to notice, correctly, that the OP based the article
> on the WP source (the fact that there is only one reference
> notwithstanding).
Well, I haven't contribute many entries so far, but wherever there is a corresponding entry in wiki, I added it to references. I think that references are the right place for the pointers to places where readers could find further informations - and online references are in a certain sense better.
In this case, I've really read the wiki entry before contributing and the formulation the basic definition is almost identical. Apart from this definition I've added my own comments. But I think that the notion of embedding in topology is so familiar, that I believe the same or analogous definition could be find in nearly all topological textbooks. I simply grabbed the first resource which was easy accessible when sitting in front of my computer.

I would appreciate if someone explained to what extent wiki-articles can be used if I don't want to violate GFDL. (drini askes the same thing in his post)

### Re: bi-directional updating

Simply put: I don't think Wikipedia is conforming to the
GFDL's requirements for acknowledging GFDL'ed source
materials, and I also think that they have set up their site
in such a way that it is infeasible to conform to the GFDL
when using their materials.

With more detail, here are the things that Wikipedia
would need to do to be compliant & use PlanetMath content
(as of last time I checked, which was in April -- I'm
assuming the situation hasn't changed, because I brought these
issues to the attention of a WP admin, who reacted fairly
dismissively). Some of these items would require action
on PM's part. And of course, if PM uses WP content, they
must do symmetrical things. However, I think PM is closer
to being compliant than WP is.

Basic steps for inclusion of modified works
(e.g. modified contents of PM on WP).

YOU MAY COPY AND DISTRIBUTE A MODIFIED VERSION OF AN FDL'ED SOURCE
DOCUMENT... YOU MUST DO THESE THINGS IN THE MODIFIED VERSION...

+ use a distinct title

- check, OK.

+ list 5 authors from the Source Document on a Title Page

* you need a Title Page
* you need to list at least 5 PM authors on it

+ list at least one author from the new Document on a Title Page

* pick at least one person and list them on the Title Page

+ state the name of the publisher on the Title Page

* I guess that would be Wikimedia Foundation?

+ preserve all the copyright notices of the document.

* you should probably have a section for Copyrights. PM's
"Legalese" section currently says

All pages on this site are copyrighted by their respective
authors.

Maybe Aaron can change this to something more
portable, e.g. "Copyright <dates> PlanetMath authors."

* put this in the same area as the other copyright notices

in the form shown in the FDL's Addendum.

* do this

+ Preserve the section Entitled History";

* For PM, that would be http://planetmath.org/?op=about

1. this as a subsection of WP's History called History of
PlanetMath

2. Make sure you also have a section stating at least the title,
year, new authors, and publisher of Wikipeda as given on the
Title Page.

+ Preserve network locations given in the Source Document... These
may be placed in the History" section.

* do this (http://planetmath.org)

+ Preserve sections Entitled Acknowledgements" or Dedications"

- I don't think PM has any

+ Preserve Invariant Sections of the Document

- I don't think PM has any

+ Delete any section Entitled Endorsements".

- I don't think PM has any

+ Preserve any Warranty Disclaimers.

- I don't think PM has any

### Re: bi-directional updating

In my view, none of the articles can be copied (using
them as references is of course OK, the same way you would
use non-FDL'ed works) until further work has been done
by the admins of both sites. When the appropriate steps
WRT licensing have been taken, then all of the wikipedia
articles should be usable.

### Tensor products

Suppose R is a DVR with field of fractions K; L is a finite separable extension of K, S is the integral closure of R in L. Suppose further that S = R[u]. Let R_hat be the completion of R with respect to its prime.

Why is R_hat[u] = S tensor-over-R R_hat? There is an obvious map
u -> u tensor 1 that extends to a map of rings; is it in fact obvious that this is an isomorphism of rings?

### Re: bi-directional updating

> must do symmetrical things. However, I think PM is closer
> to being compliant than WP is.
> YOU MAY COPY AND DISTRIBUTE A MODIFIED VERSION OF AN FDL'ED
> SOURCE
> DOCUMENT... YOU MUST DO THESE THINGS IN THE MODIFIED
> VERSION...
>
A.
> + use a distinct title
Not needed if permission is granted. However I think so far wikipdia is more careful preservin edit history than planetmath
> - check, OK.

B.
> list 5 authors from the Source Document on a Title Page
> * you need a Title Page
I don't get it, what's a title page on planetmath and what on wikipedia?

> * you need to list at least 5 PM authors on it
> + list at least one author from the new Document on a Title Page

Yes I see this point, but I entries taken the other way (wp to pm) need to list also 5 authors from WP (or all of them if they're fewer).

I'm confused still, a title page is the article itself?
> * pick at least one person and list them on the Title Page

C.
> state the name of the publisher on the Title Page
* I guess that would be Wikimedia Foundation?
I suppose, what's publisher of planetmath?

D
> + preserve all the copyright notices of the document.
> * you should probably have a section for Copyrights. PM's
> "Legalese" section currently says
> All pages on this site are copyrighted by their respective
> authors.
>
> Maybe Aaron can change this to something more
> portable, e.g. "Copyright <dates> PlanetMath authors."

but I assume you need to state the copyright notices from the originating entry to the copied one, am I right?

E
> * put this in the same area as the other copyright notices

F
> + include, immediately after the copyright notices, a
> in the form shown in the FDL's Addendum.
> * do this
Wikipedia has copyright notices at the bottom of each page
I see no copyright notice at http://planetmath.org/encyclopedia/Embedding4.html
other than the link to the legalese page
>

I.
> + Preserve the section Entitled History";
>
> * For PM, that would be http://planetmath.org/?op=about
>
> 1. this as a subsection of WP's History called History of
> PlanetMath
>
> 2. Make sure you also have a section stating at least the
> title,
> year, new authors, and publisher of Wikipeda as given on the
> Title Page.
That page refers to the story of planetmath as a site, perhaps it refers to the history of article versions? I appreciate if you could elaborate on this point

J.
> + Preserve network locations given in the Source Document...
> These
> may be placed in the History" section.
>
> * do this (http://planetmath.org)
wikipedia's entries taken from planetmath have a link back to planetmath (and to the specific entry)

K.
> + Preserve sections Entitled Acknowledgements" or
> Dedications"
>
> - I don't think PM has any
This is not the same as references section, right?

>
> + Preserve Invariant Sections of the Document
>
> - I don't think PM has any
I don't think WP have too

>
> + Delete any section Entitled Endorsements".
>
> - I don't think PM has any
I think WP neither

>
> + Preserve any Warranty Disclaimers.
>
> - I don't think PM has any

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: bi-directional updating

As I said I requested comments also from the other side
http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Plan...

let's wait
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: bi-directional updating

> > + use a distinct title
> Not needed if permission is granted. However I think so far
> wikipdia is more careful preservin edit history than

I'm simply talking about requirements of the FDL! If there
are other legally binding documents superimposed over the
FDL that recind some of its requirements, then we're in a different
situation. As it stands, WP and PM DO have different titles,
so this requirement is already taken care of.

> I don't get it, what's a title page on planetmath and what
> on wikipedia?

There isn't one on either, AFAIK. The HOMEPAGE would be a good
(or at least, pretty good) bet for each one to use, however.

> > * you need to list at least 5 PM authors on it
> > + list at least one author from the new Document on a
> > Title Page
>
> Yes I see this point, but I entries taken the other way (wp
> to pm) need to list also 5 authors from WP (or all of them
> if they're fewer).

True -- remember, my recommendations were written from the point
of view of someone telling WP what they have to do to be compliant.

> I'm confused still, a title page is the article itself?

No, there should be one title page for each DOCUMENT -- the documents,
in this case, are the PM and WP webpages. If you look closely, you'll
see that each PM article is NOT "released as a document under the FDL"
but rather, is reased as *part of* and FDL'ed document -- namely PM itself.
This is the case with WP, too (at least, I think so!).

> I suppose, what's publisher of planetmath?

A lawyer would know for sure, but I assume PlanetMath.org, Ltd., (the
PM nonprofit) would be the publisher of PlanetMath (the website).

> but I assume you need to state the copyright notices from
> the originating entry to the copied one, am I right?

Yes.

> > + include, immediately after the copyright notices, a
> > in the form shown in the FDL's Addendum.
> > * do this
> Wikipedia has copyright notices at the bottom of each page
> I see no copyright notice at
> http://planetmath.org/encyclopedia/Embedding4.html
> other than the link to the legalese page

See above -- unless EACH ARTICLE is released independently
under the FDL (which might be the case for WP, at least, perhaps
that was their intent -- but this is a separate case from the one
I'm assuming here, namely that WP is released as exactly ONE GFDL'ed
document -- and anyway, this is definitely NOT the case for PM, where
it is pretty clear that there is ONE GFDL'ed document)...then only
ONE (main) copyright page is required.

> > 2. Make sure you also have a section stating at least the
> > title,
> > year, new authors, and publisher of Wikipeda as given on
> > the
> > Title Page.
> That page refers to the story of planetmath as a site,
> perhaps it refers to the history of article versions? I
> appreciate if you could elaborate on this point

Page histories are distinct from document histories. Thus,
article versions should have no relationship whatsoever to
document versions -- unless a new version of an article
introduces a new GFDL dependency, in which case, it is
the (derivative) document's document-wide pages that must change.

> > + Preserve network locations given in the Source
> > Document...
> > These
> > may be placed in the History" section.
> >
> > * do this (http://planetmath.org)
> wikipedia's entries taken from planetmath have a link back
> to planetmath (and to the specific entry)

This is a requirement for HISTORY PAGES, which, as I stated
above, are a requirements for WHOLE DOCUMENTS. If each
individual WP page is a document, then what they have
now is *almost* sufficient, but they would need to reformat
their pages for it to be fully sufficient.

> > + Preserve sections Entitled Acknowledgements" or
> > Dedications"
> >
> > - I don't think PM has any
> This is not the same as references section, right?

Correct - not the same.

> get more feedback

Good.

### Re: bi-directional updating

you seem to have quoted the same passage twice, what did you
mean for the second quote?

### Re: bi-directional updating

I meant I got things mixed up, I'll fix that
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: bi-directional updating

> > > + use a distinct title
> > Not needed if permission is granted. However I think so
> > far
> > wikipdia is more careful preservin edit history than
>
> I'm simply talking about requirements of the FDL! If there
> are other legally binding documents superimposed over the
> FDL that recind some of its requirements, then we're in a
> different
> situation. As it stands, WP and PM DO have different
> titles,
> so this requirement is already taken care of.

yes I'm just pointing that the need of different titles can be obviated under permission, as the very fdl states:
A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.

>
> > I don't get it, what's a title page on planetmath and what
> > on wikipedia?
>
> There isn't one on either, AFAIK. The HOMEPAGE would be a
> good
> (or at least, pretty good) bet for each one to use, however.

ahh thanks :)

>
> > > * you need to list at least 5 PM authors on it
> > > + list at least one author from the new Document on a
> > > Title Page
> >
> > Yes I see this point, but I entries taken the other way
> > (wp
> > to pm) need to list also 5 authors from WP (or all of them
> > if they're fewer).
>
> True -- remember, my recommendations were written from the
> point
> of view of someone telling WP what they have to do to be
> compliant.

But.. pm imported entries would have to fulfill this too, right?

>
> > I'm confused still, a title page is the article itself?
>
> No, there should be one title page for each DOCUMENT -- the
> documents,
> in this case, are the PM and WP webpages. If you look
> closely, you'll
> see that each PM article is NOT "released as a document
> under the FDL"
> but rather, is reased as *part of* and FDL'ed document --
> namely PM itself.
> This is the case with WP, too (at least, I think so!).

I see (as you mentioned above, main pages coud serve as title pages)

(...)

> > > + include, immediately after the copyright notices, a
> > > in the form shown in the FDL's Addendum.
> > > * do this
> > Wikipedia has copyright notices at the bottom of each page
> > I see no copyright notice at
> > http://planetmath.org/encyclopedia/Embedding4.html
> > other than the link to the legalese page
>
> See above -- unless EACH ARTICLE is released independently
> under the FDL (which might be the case for WP, at least,
> perhaps
> that was their intent -- but this is a separate case from
> the one
> I'm assuming here, namely that WP is released as exactly ONE
> GFDL'ed
> document -- and anyway, this is definitely NOT the case for
> PM, where
> it is pretty clear that there is ONE GFDL'ed
> document)...then only
> ONE (main) copyright page is required.

point taken

> > > 2. Make sure you also have a section stating at least
> > > the
> > > title,
> > > year, new authors, and publisher of Wikipeda as given on
> > > the
> > > Title Page.
> > That page refers to the story of planetmath as a site,
> > perhaps it refers to the history of article versions? I
> > appreciate if you could elaborate on this point
>
> Page histories are distinct from document histories. Thus,
> article versions should have no relationship whatsoever to
> document versions -- unless a new version of an article
> introduces a new GFDL dependency, in which case, it is
> the (derivative) document's document-wide pages that must
> change.

ok, what would you think wikipedia should do regarding this point?

> > > + Preserve network locations given in the Source
> > > Document...
> > > These
> > > may be placed in the History" section.
> > >
> > > * do this (http://planetmath.org)
> > wikipedia's entries taken from planetmath have a link back
> > to planetmath (and to the specific entry)
>
> This is a requirement for HISTORY PAGES, which, as I stated
> above, are a requirements for WHOLE DOCUMENTS. If each
> individual WP page is a document, then what they have
> now is *almost* sufficient, but they would need to reformat
> their pages for it to be fully sufficient.

these *may* be placed in the history section. WP places on the document iself (and also preserves the notice on all previous versions, linked from the history dialog)

(...)

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

### Re: bi-directional updating

> > > > + use a distinct title
[...]
> yes I'm just pointing that the need of different titles can
> be obviated under permission, as the very fdl states:
> A. Use in the Title Page (and on the covers, if any) a title
> distinct from that of the Document, and from those of
> previous versions (which should, if there were any, be
> listed in the History section of the Document). You may use
> the same title as a previous version if the original
> publisher of that version gives permission.

OK, but then they'd need to give permission, obviously.

> > > I don't get it, what's a title page on planetmath and what on
> > > wikipedia?
> > There isn't one on either, AFAIK.
[...]

I think the key would simply be to run through the GFDL and state how
the document in question is to be interpreted for GFDL compliance
purposes. If this statement says that such-and-such a page is the
"title page" and that page has all the properties expected of it under
the GFDL, then it can serve as a title page.

> >
> > > > * you need to list at least 5 PM authors on it
> > > > + list at least one author from the new Document on a
> > > > Title Page
[...]
> But.. pm imported entries would have to fulfill this too,
> right?

Sure, of course, but in reverse; PM would have to list 5 WP authors.

> ok, what would you think wikipedia should do regarding this
> point?

be done.

The first step (IMO) would be to clarify whether WP is ONE document or
MANY. If it is MANY, then a record of PM's contribution (phrased as
per the GFDL's requirements) will be required in the history section
for each wP page that uses PM content. Otherwise, if it is ONE, then
only one such record would be required.

I don't know how WP wants to be interpreted, legally speaking, and I
don't know if compliance is really all that important to them. But if
they care about it, they should be able to go through the GFDL item by
item and say how they satisfy that item (not dismiss it as unimportant
or whatever). So should PM.

Sorry to seem to give an elusive answer, but I think a step-by-step
approach would give the best answers; if my opinion is desired at some
point in a process like this, just ask.