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)