Let be the family of sets in such that is empy or is finite.
Show that is a topology on . By considering the inverse images of closed sets, show that every polynomial function is -continuous.
Hey everyone, I was wondering if anyone could assist me with the 2nd part. If is closed in then is the empty set, or A is finite, i.e. A is or finite.
I don't understand though how is closed and I would really appreciate some advice here, thank you so much. This would show continuity of the polynomial functions.
it can have at most the degree of the polynomial am I right? eg a polynomial of degree three has 3 complex roots so at most 3 real roots.
so then the number of elements in . Is this correct?
i.e. Hence is finite!
But what about if A = ? Then I guess I am trying to show that . I guess if f is a polynomial it just moves every point in R to some other point in R. i.e. take any point in R and the polynomial will plop out another point, so it is its own image.
Thanks for the help
This is called the cofinite topology on . Notice that the resulting topological space is . And so let be a polynomial, and let be closed, then since it is the compliment of an open set we have that is finite. Let . Then clearly for each we have that (otherwise the polynomial would be a polynomial of degree with more than roots! Thus, we see that . Thus, is finite and thus by ness closed.
The conclusion follows.