Does there Exist a Continuous Map ?

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The discussion centers on whether a continuous map exists between any two topological spaces (X,T) and (X',T'). The argument suggests that by using the Axiom of Choice, one could define a function f such that the preimage of every open set U in T' corresponds to an open set V in T. This approach implies the potential for establishing a well-defined continuous map between the spaces. Additionally, the concept is extended to measurable spaces, questioning if a similar method applies. The inquiry remains open regarding the validity of this mapping technique.
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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.
 
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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.
 

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