For #6, I have what looks to me like a counterexample. Possibly I am using the wrong definition of continuous function? I am taking this one as my source.
Take as the topological space X the real line under the Euclidean topology. Let f′(x,y) be the point in S1 at (x+y) radians rotation. This is surjective; all points in S1 are mapped to infinitely many times. It is also continuous; For every (x,y)∈X,X and neighborhood N(f′(x,y)) there is a neighborhood M(x,y) such that ∀p∈M, f(p)∈N
The partial functions f(x) from X→S1 are also continuous by the same argument.
Currying doesn’t preserve surjectivity. As a simple example, you can easily find a surjective function 2×2→2, but there are no surjective functions 2→22.
For #6, I have what looks to me like a counterexample. Possibly I am using the wrong definition of continuous function? I am taking this one as my source.
Take as the topological space X the real line under the Euclidean topology. Let f′(x,y) be the point in S1 at (x+y) radians rotation. This is surjective; all points in S1 are mapped to infinitely many times. It is also continuous; For every (x,y)∈X,X and neighborhood N(f′(x,y)) there is a neighborhood M(x,y) such that ∀p∈M, f(p)∈N
The partial functions f(x) from X→S1 are also continuous by the same argument.
Currying doesn’t preserve surjectivity. As a simple example, you can easily find a surjective function 2×2→2, but there are no surjective functions 2→22.
Ah, yes, that makes sense. I got distracted by the definition of Cont(A,B)’s topology
and applied it to surjectivity as well as continuity.