# Does there Exist a Continuous Map ?

1. Nov 25, 2011

### Bacle2

Hi, All:
I saw this question somewhere else: we are given any two topological spaces (X,T), (X',T'), and we want to see if there is always at least one continuous map between the two. The idea to say yes is this: we only need to find f so that f-1(U)=V , for every U in T', and some V in T. So, it seems , for the infinite case, we could use choice to assign to each U in T' some V in T. Does this give us a well-defined map between (X,T) and (X',T')? It seems like we could then , similarly, define a measurable map between two measurable spaces using the same idea. Does this work?
Thanks.

2. Nov 25, 2011

### Bacle2

Just to clarify: I know if X' as above is not empty, then any constant map is continuous; I was just curious to see if the setup above worked.