Class-metric

In two weeks I start lecturing Topology I at our University. I am excited. This course was my favourite among the basic undergraduate courses. The course concentrates on metric topology and its goal is to prove simple results about complete and compact spaces
such as the Banach Fixed Point Theorem.

Hence, I am preparing some examples and exercises about metric spaces… and I share one with you:

Exercise: Let $$\mathbb{F}$$ be the class of all finite sets. For every finite sets $$x$$ and $$y$$ let $$d(x,y)=\#(x\triangle y)$$, where $$x\triangle y=(x\setminus y)\cup (y\setminus x)$$ is the symmetric difference and $$\#X$$ is the cardinality of $$X$$. Does $$d$$ define a metric on $$\mathbb{F}$$?

A metric on a class? you say. Well, call $$d$$ a class-metric, if you like!

…and because I am a set theorist:

Fact: Let $$X$$ be any uncountable set (or class) of finite sets. Then there exists an uncountable subset $$Y\subset X$$ such that $$d(x,y)=d(x’,y’)$$ for all $$x,y,x’,y’\in Y$$.

Cool?