Does there Exist a Continuous Map ?

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SUMMARY

The discussion addresses the existence of continuous maps between any two topological spaces (X,T) and (X',T'). It establishes that a continuous map can be defined by finding a function f such that f-1(U) = V for every open set U in T' and some open set V in T. The use of the Axiom of Choice is suggested for infinite cases to assign V for each U, leading to a well-defined map. Additionally, the concept of defining measurable maps between measurable spaces using a similar approach is introduced.

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  • Understanding of topological spaces and their properties
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Bacle2
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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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