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.
Originally Posted by slevvio
Hint: if f(x) is a polynomial of degree > 0, how many solutions at most can the equation , for fixed k, have?
So now, how many elements can the set have, when A is finite?
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