Definition talk:Topological Space

From ProofWiki
Jump to navigation Jump to search

Can a topological space be empty?

Much of the work in the topology category takes for granted that if $T = \left({S, \tau}\right)$ is a topology then $S$ is non-empty. A trivial topological space is defined as a topological space whose underlying set is a singleton.

However, some recent amendments to some of the results in this category have been an extra condition added to define specifically that a certain property is expected to apply when (using above notation) $S$ is non-empty, that is, assuming the existence of an empty topological space.

In none the source works I have to hand (except one) is raised the possibility that $S$ is empty. It appears tacitly assumed that a topological space always contains at least one element. The exception is 1964: Steven A. Gaal: Point Set Topology who, after defining a topology by means of the open set axioms, drops into the discussion of his definitions: "We shall often use the expression $X$ is a topological space. This means that $X$ is a nonvoid set and a topology $\mathscr T$ is given on $X$." (My emphasis.) And then again: "It is possible to define a topological space on any nonvoid set $X$ ..."

So what? It is also possible (and uniquely) to introduce a topology on the empty set. Perhaps the less obvious claim was stated, with the understanding that the other one (the more trivial one) every reader can see for themselves instantly. Oh, well, it really doesn't matter. Wlod (talk) 02:13, 23 December 2012 (UTC)

This is the only reference I can find on this point, but I think it's worth thinking about: do we:

(a) Include in the definition of a topological space $\left({S, \tau}\right)$ that $S$ should be non-empty, thereby deliberately excluding a particularly degenerate case


(b) In every page where we establish a result, specifically take into consideration whether $S$ is empty or not, thereby adding clutter to the result in question but ensuring that the degenerate case above is accounted for?
The opposite is true. You force much more clutter and real complications by excluding the empty space (suddenly, before applying the apparatus of topology, one has to prove that a space is non-empty, despite the fact that most of the topological technique smoothly applies to the empty space as well). Wlod (talk) 02:13, 23 December 2012 (UTC)
What we need on this site are contributions towards some of the areas where we are lacking considerably. We have pretty well established certain areas of mathematics, and have taken a view on how to present certain constructs according to the material which has been available. If, as a result of the development of further work it proves necessary to provide alternative definition strategies, then such viewpoints may end up being taken more seriously. Until that time, the original specifications apply according to the sources available to the authors. --prime mover (talk) 07:41, 23 December 2012 (UTC)
the sources available to the authors--means nothing. Also, a piece of printed junk available to authors should not prevail over a comment which represents mathematical objective reality.
If you are not familiar with Bourbaki, Kelley, Engelking, ... then why do you bother with this topic (general topology)? These three are classical. Dugundji is very strong too (adds quite a bit of more geometric topology). One of the first was Kuratowski. However, in those early days, in the first (French) edition he considered only T1-spaces. Later, certainly in English editions, all spaces, including the empty space, of course. The absolute classic on (topological) Dimension Theory is Hurewicz & Wallman. Check also Nagata. They wouldn't dream about not including the empty topological space. One has to take into account also algebraic topology, starting with Eilenberg & Steenrod. Are you going to destroy their beautiful monography, on which generations of topologists were raised? Are you going to make it obsolete? :-) Oh, it's not available to authors? :-)
And still, the most important are mathematical ear and mathematical (or universal) taste. When no solution is superior, fine, one may chose any of them, or even more than one. Whatever the choice, it has to be executed cleanly, e.g. without garbage axioms (as in the other topic). Put your hand on your heart, and ask yourself, and honestly answer the question about your ear and taste. If you're not confident then don't insist on your suggestions. Propose them but don't be stubborn.
Even classical solutions can be unnecessarily risky, etc. E.g. Bourbaki, against common sense, forces its convention of set-intersection of the empty family. A common sense solution is that, if you allow it at all (preferably one does), then it should be the class (not set) of all sets. But Bourbaki insists on having a local universal set relative to the given conditions. Perhaps they could pull it off, who knows, but it's way to tricky (prone to errors), and unrealistically hard to implement in everyday mathematical practice. Their intersection axiom states that intersection of arbitrary finite family of open sets is open. Thanks to their convention, it follows that the empty set is open--one does not need a special axiom for this, which makes Bourbaki happy. I prefer a much more robust situation, where the intersection of the empty family is always the class of all sets, and then one still can be elegant enough by formulating the intersection axiom for space $(X\ \tau)$ as follows:
$X \cap \Cap K$   is open for every finite family $K$ of open sets.
(My axiom works even when you do not allow intersections of empty families at all(!), but instead we assume a notational convention: when an intersection happens to be of an empty family (often we can't tell) then it's not there--it's like there is nothing on paper or screen; this is possible only when such an intersection is intersected still with another set. This notational approach is not my first choice--I mention it just to have it on record, for the discussion sake).
Sorry for a long comment. I was concrete, specific. We had a bunch of muddy, arbitrary comments around, on which a mathematician just should not waste their time, should just pack and go away from this place. (Hmmmm... ).
Wlod (talk) 05:17, 26 December 2012 (UTC)
the sources available to the authors means "the books currently on my bookshelf". No, I am not familiar with Bourbaki, Kelley, Engelking.
As for the rest of your post, it's all argumentum ad verecundiam anyway.
If your suggestion is that only people who have read your specific sources should be allowed to contribute to a mathematics wiki then feel free to apply to be an administrator of this site so you can selectively block whoever you feel is unworthy. They you will be able to change everything to suit your fascist whim. --prime mover (talk) 09:58, 26 December 2012 (UTC)

My vote is for (a), as the prospect of implementing (b) decreases my morale. Besides, this is the approach taken by all the works I have seen on this subject, but for the fact that (apart from Gaal) they omit to mention it. --prime mover (talk) 11:05, 2 December 2012 (UTC)

Vote: Option (a)
We may vote that 2+2=5 but it won't make it so. Wlod (talk) 02:13, 23 December 2012 (UTC)
$2.4 + 2.4 = 4.8$ which when you round to the nearest integer, $2+2=5$. $\blacksquare$ --prime mover (talk) 07:33, 23 December 2012 (UTC)
Authority: Professor Efe A. Ok of NYU
External Links:
Section of interest
General Page
He is yet to publish his book Elements of Order Theory so it may not be worthy but his CV demonstrates he is active in the field.
--Jshflynn (talk) 21:10, 2 December 2012 (UTC)

Comment on decision

The decision about disallowing the empty topological space is harmful. Wlod (talk) 03:11, 22 December 2012 (UTC)

Any argument whatsoever?! You cannot truly be expecting that we would reconsider anything without giving at least some reason. In fact, your comment could be understood as downright arrogant - as were you the harbinger of the mathematical society, visiting some illiterate baboons (still trying to invent ways to count their bananas) to try and enlighten them. --Lord_Farin (talk) 09:26, 22 December 2012 (UTC)
+1FTW --prime mover (talk) 09:32, 22 December 2012 (UTC)
There is no reason to exclude the empty space in the first place. Topology and mathematics needs both: topological categories with the empty space, and, less often, without. Sometimes we work with the pointed spaces, which automatically are non-empty, but that's not a straight topological category but a category of pairs (a subcategory of all pairs of spaces). (Ironically, a topic of my own requires the category of non-empty topological spaces). But all this does not mean that one should delete the category of all spaces, including the empty one, when we can have both.
In algebraic topology the homology and cohomology groups are applied to spaces, which often are subspaces. These groups are not applied to subsets but to spaces. And these spaces often have to be empty, as in the long homological sequences.
There are situations, quite common, when we deal with a family of (indexed) spaces, and we have no idea whether or not some of them are empty. Then all proofs would get complicated. This need of extremal (even denerated) objects is a common situation in mathematics.
One should not rely on a random textbook. One should refer to the most authoritative ones. But even more importantly, one needs to follow the natural flow of mathematics. Most of the time it provides a natural solution.
Wlod (talk) 23:45, 22 December 2012 (UTC)
Who is to determine whether a textbook is "random" or "authoritative"? 1964: Steven A. Gaal: Point Set Topology is cited quite a lot, which to me indicates that it can be considered "authoritative". I admit that the only reason I've used 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces is because it was the set work for my own course of studies many years back, and as such is pretty "random" (although it appears on plenty of set texts in undergraduate topology courses in the UK) - and that work also specifically states that the underlying set be non-null. 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) is quiet on this point, but it needs to be pointed out that it is considered by many (for better or worse) to be, if not authoritative, then at least influential. As for 1967: George McCarty: Topology: An Introduction with Application to Topological Groups, similar lack of specification, but it needs to be pointed out that its emphasis is towards topological groups, and so is predicated upon the underlying set being a group which is by definition non-empty. (As for the appallingly badly-written Hocking and Young, I have not seen fit to consult it, as it got lost in transit between continents and I never bothered to replace it.)
In consequence, cogent though your argument might be, as I have seen none of your so-called "authoritative sources" (you never specified which you favour), I remain unconvinced. The approach taken by this website: that it is acknowledged that there is a split between exclusionists and inclusionists, and that we take the exclusionist approach (as it coincides with the considerable work which has already been posted up on this site), will stand at present - at least until we see some specific need to the contrary. --prime mover (talk) 07:32, 23 December 2012 (UTC)

The inductive definitions of the topological dimension invariably start with the initial condition:

the dimension of the empty space is -1, $\dim(\empty) := -1$.

Wlod (talk) 00:01, 23 December 2012 (UTC)

Thank you for bringing in solid arguments. They've convinced me - apparently both including and excluding the empty space bring certain awkward situations, and it seems more natural to talk about "non-empty" than about " or empty set". --Lord_Farin (talk) 00:18, 23 December 2012 (UTC)
Well thought and well said! Wlod (talk) 02:20, 23 December 2012 (UTC)
As and when such definitions become necessary according to the nature of the material posted up which depends on it, we will consider such a change to our policy. --prime mover (talk) 15:24, 23 December 2012 (UTC)
Yes, I'd agree to that. We'd be very pleased to see you covering some source works (or at least mentioning them or write up a page for them on PW:Books) since that actually brings us somewhere; also, leading by example often works to clarify ideas and trains of thought for those that otherwise may not follow. For now, the immensity of the task to change all topology pages to accommodate this change of "policy" justifies PMs call for more concrete material supporting your claims (which, as they are (probably) valid, shouldn't be too hard but for finding time in your schedule to do so). I'm sure you understand. --Lord_Farin (talk) 22:16, 23 December 2012 (UTC)
As it always was and is now (regardless of accidental hiccups) we were able to talk smoothly about topological spaces and operations on the (families) of topological spaces. Do you really want to rewrite topology in the following awfully clumsy way:
  • consider a family of topological spaces, possibly some of them not actually spaces but the empty set, ...
  • consider the topological union of a family of spaces and possibly the empty set (possibly repeated many times). The union is always a topological space or the empty set.
  • etc.
As you have observed yourself, it is simpler to say, when needed, consider the Cartesian product of non-empty topological spaces, rather than on other occasions consider the Cartesian product of topological spaces and possibly the empty set (repeated many times).
When we make the non-empty assumption then there is an organic reason for this, and the phrase adequately provides information. There is no artificial complication. But if you start to write about families of topological spaces or empty sets, then this awkwardness will be forced artificially by an arbitrary (not organic) nonsensical decision.
Wlod (talk) 05:35, 26 December 2012 (UTC)
As I intended to convey by means of my comment, I think there should be a prolonged discussion surpassing personal preference on the matter before any amendments to the whole of the rather comprehensive topology section are made. All I am asking for is some more-or-less independent authority that makes clear why allowing the empty set to be a topological space is better (except for reasons of brevity, which in my book would suffice). Some entities on this wiki (justifiably) refuse to accept this convention solely on the (IMO quite convincing) plead you gave for it. --Lord_Farin (talk) 13:43, 26 December 2012 (UTC)
Let me add to that my view on the matter, or indeed any matter where possibly the readable but possibly sloppy alternative and the more formal though cumbersome alternative are opponents. Readability, if necessary amended with explanatory notes, is IMHO to prevail over formal yet cumbersome accuracy. It is evident that the formulations as sketched by Wlod are obfuscating what is to be conveyed. This makes it most unpleasant and deterring to read proofs and/or use definitions. In other cases, the balance naturally sways to the formal side (e.g. $\left({S, \tau}\right)$ over $S$) and I think that while there is some grey area in between these two extremes, we should keep in mind that ultimately this stuff is to be read by interested or searching visitors that sometimes do not wish to be tired with decisions the wiki editor base has made that, while maybe formally more accurate, make the material unreadable. Such will inevitably, in the long run and if done sufficiently many times, make ProofWiki irrelevant to the point where it becomes a narcissistic, useless exercise (and indeed I think such is not the case at the moment).
Concretely, I thus say that "non-empty topological space" may be enough pedantry for our readers, and "topological space or empty set" brings too much trouble with it (such as a never-ending road of tire- and cumbersome rewriting of present and future articles in the topology section). --Lord_Farin (talk) 13:43, 26 December 2012 (UTC)

As I've mentioned before, theorems about product spaces without assuming AoC get rather more awkward, because the product of underlying sets of a collection of (non-empty) spaces may be empty, so that in fact the product of topological spaces may not exist.

  • Tychonoff's theorem is known to be equivalent to AoC, but to prevent this equivalence from being a triviality, and to make the theorem sensible in this context, the theorem must be rewritten from its usual form to say something like this: Suppose the product of the underlying sets of a collection of compact topological spaces is non-empty. Then the product of the spaces is compact.
  • As a simpler matter, the fact that products of connected spaces are connected must be rewritten: Suppose the product of the underlying sets of a collection of connected topological spaces is non-empty. Then the product of the spaces is connected. --Dfeuer (talk) 18:07, 30 December 2012 (UTC)
Dfeuer (please sign your post btw, I'd do it for you now but I have stuff to do) - not sure which side of this debate you're coming down on. Do I take it you're exclusionist? --prime mover (talk) 09:46, 26 December 2012 (UTC)
I'm an inclusionist. Actually, I've just run into another spot where excluding empty spaces makes an annoying problem: the Stone space of a non-empty topological space may be empty if one does not assume the ultrafilter lemma, This probably isn't a huge problem, since Stone's Representation Theorem for Boolean Algebras depends on BPI, but it's irritating. I really don't see what you have against the space $(\varnothing,\{\varnothing\})$ --Dfeuer (talk) 18:07, 30 December 2012 (UTC)

Returning to this after a couple of years in other areas ...

I am going to experiment with removing the stipulation that a topological space be non-empty. Bear with me -- I'm just going to see where this goes to, and whether there are any serious negative effects to this. Sorry, but I learn all the time that I am wronger than I thought I was. --prime mover (talk) 20:57, 3 May 2015 (UTC)