Definition talk:Topological Space

Can a topological space be empty?

Much of the work in the topology category takes for granted that if $T = \struct {S, \tau}$ 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 $X$ ..."

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)