Definition talk:T5 Space/Definition 2

We have a flag: {{mistake|If we demand $A^- \subseteq Y$ a priori then certainly $Y$ contains a closed nbhd of $A$}}

This is exactly how the condition is worded in 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): $\text{I}: \ \S 2$.

Is it actually a "mistake" as such, then? Or is it indeed accurate despite the a priori condition? --prime mover (talk) 10:10, 25 November 2012 (UTC)

I have written out what it means, and it seems that definition 2 is implied by, but does not imply definition 1. The latter is seen as follows:

Given $A^- \cap B = A \cap B^- = \varnothing$, the only reasonable thing to do is to take $Y = B^c$, yielding the premises of def 2. Thus we find $A'$ closed with $A \subseteq A' \subseteq Y$, and juggling with complements yields $B \subseteq (A')^c, A \subseteq Y^\circ$ and so $(A')^c, Y^\circ$ as the only natural $U,V$. It remains to verify they are disjoint, but $(A')^c \cap Y^\circ = \varnothing \iff Y^\circ \subseteq A'$. This condition cannot be derived (take e.g. $A = A' = \varnothing$) and so def 2 doesn't imply def 1.

I haven't managed to adjust def 2 to be equivalent to def 1. The intuitive repairing condition (demanding $Y^\circ \subseteq A'$ possible) comes down to asking, for all $Y$, $(Y^\circ)^- \subseteq Y$, but I haven't checked this against def 1; it may be too strong. --Lord_Farin (talk) 13:29, 25 November 2012 (UTC)

That def 2 couldn't be right is immediate from the fact that its content is vacuously true... --Lord_Farin (talk) 13:30, 25 November 2012 (UTC)

Suppose $A \subseteq Y$. In order for $A^-$ to be a neighborhood of $A$, there needs to exist $U \in \tau$ such that $A \subseteq U \subseteq A^-$.

Let $\R$ be the real number line under the usual topology. Let $A = \left({0, 1}\right]$. Then $A^- = \left[{0, 1}\right]$ but there is no open interval $U$ such that $A \subseteq U \subseteq A^-$. So $A^-$ is not, in this context, a closed nbhd of $A$. So if $Y = \left[{0, 1}\right]$ then while it is the case that $A^- \subseteq Y$, it is not the case that $Y$ a priori contains a closed nbhd of $A$.

Or am I missing something? --prime mover (talk) 15:06, 25 November 2012 (UTC)

Definition 2 does imply definition 1. Let $B = Y^{\complement}$. The conditions that $A \subseteq Y^\circ$ and $A^- \subseteq Y$ mean that $A$ and $B$ are separated. If $N$ is a closed neighborhood of $A$ that is contained in $Y$, then $N^{\complement}$ is an open neighborhood of $B$. We have $N \cap N^{\complement} = \varnothing$. Anything wrong? --abcxyz (talk) 17:41, 25 November 2012 (UTC)

It really does appear that both definitions are equivalent as they stand. I suggest to remove the {{mistake}} template. --abcxyz (talk) 22:09, 25 November 2012 (UTC)

$N^c$ is not an open nbhd of $A$. --Lord_Farin (talk) 22:25, 25 November 2012 (UTC)

I said it is an open neighborhood of $B$. --abcxyz (talk) 22:27, 25 November 2012 (UTC)

Sorry. I meant $N$ of course. --Lord_Farin (talk) 22:28, 25 November 2012 (UTC)

Yeah but $N$ contains an open neighborhood of $A$ (by definition). --abcxyz (talk) 22:32, 25 November 2012 (UTC)

The terminology closed neighbourhood in current use I hereby (strongly suggest to) eviscerate from the PW language. It is ridiculously misleading, since I have read it every time as meaning "closed superset" - hence the cause of confusion. Thanks for shining light on this, though. Why would people even invent such an apparent cause of confusion?! --Lord_Farin (talk) 22:38, 25 November 2012 (UTC)

How is it confusing? A "closed neighborhood" of $A$ exactly means a neighborhood of $A$ that is a closed set of $T$. --abcxyz (talk) 22:41, 25 November 2012 (UTC)

I thank you for your cooperative mindset. Like I would actually go through so much effort were it not for actual confusion to occur. Apparently our subconscious paradigms differ. I still contend that the terminology is compromised. IMHO, the word neighborhood itself should be avoided, unless prefixed with 'open'. --Lord_Farin (talk) 22:46, 25 November 2012 (UTC)

LF, I'm with abcxyz on this one. A neighborhood of $A$ is a superset which contains an open set which in turn contains $A$. That is what it is. A closed neighborhood is one of those, which happens to be closed.

If a closed neighborhood is the same thing as a closed superset, then there is no point of the precisely-defined concept "closed neighborhood" in the first place.

For motivation, note that the various separation axioms would not be definable without the concept of "neighborhoods" and "closed neighborhoods".

abcxyz: unless you get to it before me, I'll post up and formalise your above proof in due course. --prime mover (talk) 22:47, 25 November 2012 (UTC)

Fair enough. I'll simply refrain from interfering with pages bearing the word 'neighborhood' in the future. --Lord_Farin (talk) 22:48, 25 November 2012 (UTC)

There is a note in the "neighborhood" page which points out that using "neighborhood" to mean specifically "open neighborhood" is deprecated generally in the mathematical community, because that's what I have understood from my reading. Any open superset of $A$ is therefore a neighborhood of $A$ and you lose a lot of powerful context: the idea of there being an open set between $A$ and a general superset of $A$ gives you a much more compact technique to specify the various degrees of separation. --prime mover (talk) 22:53, 25 November 2012 (UTC)

Sorry about this but I'm not letting this go quite yet, I need to know whether we need to amend our page:

"... I have read [closed neighborhood] every time as meaning "closed superset"."

By this, do I understand it as meaning whenever you have encountered the definition for "Closed Neighborhood" it has always been defined just as "closed superset"? If so, then those sources need to be exposed and we need to comment in an "also defined as", otherwise the confusion will be propagated. --prime mover (talk) 07:40, 26 November 2012 (UTC)

That is correct. However, none of my topology courses had an actual book, so the best that can be provided is course notes (and I know how you feel about those). It may also be that it was just my brain having a short circuit (one that I wasn't already aware of :) ). I do think however that such a note is a good idea. --Lord_Farin (talk) 09:39, 26 November 2012 (UTC)

... as long as we can establish that a "closed neighborhood" is sometimes "correctly" described as the same thing as a "closed superset", of which, I'm sorry, I doubt. If your course notes are the only source of info you have stating this, then I expect they are incorrect. All the books I have, and every web resource I've seen, insist on the "open set" condition as described above.

If we were to add a note, it would be along the lines of: "beware, a closed superset is not necessarily a neighborhood" and show an example, e.g. $[a..b]$ is not a neighborhood of $[a..b)$. If there is a reference that can be verified that a closed neighborbood can be merely a closed superset, then we need to be all over it like a Customs dope hound to check that's really what it means before letting it through. --prime mover (talk) 13:59, 26 November 2012 (UTC)

The "beware" note is the only thing I would wish to be added. Moreover, keep in mind that not all topology texts concern themselves with separation axioms or other areas where the term neighborhood naturally/necessarily carries the 'contains open superset' condition. It is not like I was suggesting that the terminology, while having a different meaning, was used to still produce the same definition verbatim; that'd be ridiculous. --Lord_Farin (talk) 14:07, 26 November 2012 (UTC)

Whether or not neighborhoods are introduced in conjunction with separation axioms is irrelevant (although I'd be surprised at a text that does not even mention Hausdorff spaces). The fact is: calling something a "neighborhood" which does not require that there is an open set inside it somewhere is the real confusion. The unconventional and confusing approach is the one taken by your course notes. Every other approach that I have seen does it the way that ProofWiki does it. Are your course notes on line? I'd like to see them. --prime mover (talk) 17:55, 26 November 2012 (UTC)

Case settled, then. Course notes not online AFAIK; also, don't neglect the considerable mutation concepts may have undergone in my head in the years I hardly touched topological subjects. --Lord_Farin (talk) 18:02, 26 November 2012 (UTC)

To put this issue to bed, I have nowiki'd out the invocation of the Mistake tag so as not to generate false hits in that category. --prime mover (talk) 18:55, 26 November 2012 (UTC)

Incidentally, and interestingly, that has also fixed the wrong text appearing in abcxyz's comment (the {{mistake}} in there read {{Proofread}} until recently). I have looked through MediaWiki bug reports but I haven't been able to locate the origin of the behaviour; maybe I should post a new bug report on there. This issue still persists on Help:Editing. --Lord_Farin (talk) 19:05, 26 November 2012 (UTC)

Puzzling. The one at the top of the page reads Proofread but the one in the text immediately above this still says "mistake". Wonder how long that will last ... --prime mover (talk) 19:09, 26 November 2012 (UTC)

I tried to use the 'code' HTML tag rather than 'tt', which moved the problem back down again. I suspect that it says 'Proofread' because that's the last statement inside 'nowiki' tags on Template:Mistake. Sux. --Lord_Farin (talk) 19:38, 26 November 2012 (UTC)