Talk:Schröder-Bernstein theorem/Draft

"Details": "the induced induced image"? Boris Tsirelson 06:29, 26 September 2010 (UTC)

Thanks. Corrected. --Peter Schmitt 12:26, 28 September 2010 (UTC)

"Proof:Proof":

$S_{1}\subset S\subset A\Rightarrow f_{\ast }(S_{1})\subset f_{\ast }(S)$

probably should be

$S_{1}\subset S\subset A\Rightarrow \sigma _{\ast }(S_{1})\subset \sigma _{\ast }(S)$

Oops -- correct but not what is needed. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Monotone" in general may be understood as "either increasing or decreasing"; it is meant "(momotonely) increasing" or "isotone".

Yes, that was negligent. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Proof:Proof":

"By assumption, there are injective functions (...) that induce..."

I'd say

"By assumption, there are injective functions (...); they induce..."

because the second part of the phrase is not a part of the assumption (but its consequence).

True -- that is better. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Outline": the reader can guess what is denoted by f and g (or see the details), but we'd better let him know.

I forgot that I did not introduce it before. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Details": probably also (4) is needed, explaining what are A₂, B₁ and B₂ (which is easy) and why B₁ is the image of A₁ under f and A₂ is the image of B₂ under g (which is less easy).

I was lazy -- I thought this is "obvious". --Peter Schmitt 23:37, 5 October 2010 (UTC)

Boris Tsirelson 12:19, 5 October 2010 (UTC)

Done. Thanks. --Peter Schmitt 23:37, 5 October 2010 (UTC)

It seems, some new "decreasing" should rather be "increasing". Boris Tsirelson 13:18, 6 October 2010 (UTC)

Fixed. My only excuse is that it was very late and the mental image was the decreasing sequence produced by σ starting at A. --Peter Schmitt 09:14, 7 October 2010 (UTC)

More remarks

"Outline": "This defines a mapping of subsets of A to subsets of A that is monotone" — rather, increasing.

"Details": "(2) σ is a monotone function" — rather, increasing.

"Details": "$A_{2}:=A\setminus A_{2}$" — rather, $A\setminus A_{1}$.

"Details": "that has the desired properties:" — either a continuation will follow, or the colon should be a fullstop.

In addition, some commas and fullstops after displays are missing. Boris Tsirelson 17:33, 12 October 2010 (UTC)