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 $X$ ..."

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?

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)


 * 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)

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)