Let

be a ring homomorphism. Let

and

. If

, then

is a prime ideal of

, i.e., a point of

. Hence

induces a map

. Show that:

If

is surjective, then

is a homeomorphism of Y onto the closed subset

of .

[

denotes the set of all prime ideals of

which contain

].

Here is one way to do it:

By the "Ideal Correspondence Theorem," we know that and induces a bijective map from to . Then I need to show

is continuous and

is continuous to show it is a homeomorphism.

I don't understand the sentence in red. How do we know

induces a bijective map from to ?