Definition talk:Topological Space

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 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

or:
 * (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)

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"? is cited quite a lot, which to me indicates that it can be considered "authoritative". I admit that the only reason I've used  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.  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, 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 top.space" than about "top.space 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)