4

I read that the projection mapping $\pi_x: X\times Y\to X$ is continuous. For this function to be continuous, the inverse $\pi^{-1}_x: X\to X\times Y$ has to exist.

Take any point $x\in X$. The inverse $\pi^{-1}_x(x)$ has multiple values- $(x,y_1),(x,y_2)$, etc. This is a one-many mapping. How come the inverse is defined then?

Thanks in advance!

1 Answers1

4

The inverse, as a function from $X$ to $X\times Y$, is not defined (unless $Y$ is a one-point space). It is possible to define a set-valued inverse from $X$ to $\wp(X\times Y)$ that takes $x\in X$ to the set $\{x\}\times Y$, but it’s not possible to talk about continuity of this map unless one puts some topology on $\wp(X\times Y)$, the set of all subset of $X\times Y$.

Brian M. Scott
  • 616,228
  • Brian, like you said, I thought ${x}$ would be mapped to ${x}\times Y$. What does "put some topology" mean, and how woud one go about doing that here? Thanks –  Aug 30 '13 at 06:10
  • If you have a set $S$, you can talk about topologies on $S$: $\tau\subseteq\wp(S)$ is a topology on $S$ if it satisfies the usual conditions. This is true even if $S$ is a set of subsets of some topological space. It turns out that certain sets of subsets of topological spaces can be given useful topologies. The family of all non-empty closed subsets of a space, for instance, can be given the Vietoris topology, and the family of finite subsets can be given the Pixley-Roy topology. I suspect that you don’t really ... – Brian M. Scott Aug 30 '13 at 06:20
  • ... want to get involved with either of these just yet; you should probably be really comfortable with the more basic topological constructions first. – Brian M. Scott Aug 30 '13 at 06:20