In his book "Algebraic Geometry and Arithmetic Curves", Liu defines open/closed immersions of locally ringed spaces in terms of topological open/closed immersions:

enter image description here

What does he mean by the terms "topological open (resp. closed) immersion"?

Does he mean that

  1. $f(X)$ is an open (resp. closed) subset of $Y\!,\,$ and

  2. the induced map $X\to f(X); \;x \mapsto f(x)$ is a homeomorphism?

Many thanks! :)


Yes, that's a correct definition. Yours (1.) is also equivalent to 2. below.

  1. $f(X)$ is open (closed) and $f$ is a homeomorphism on its image
  2. $f$ is open (closed) and a homeomorphism on its image

If we then define an immersion to be a homeomorphism on its image, then an open (closed) immersion really is an immersion that is open (closed).

Note: it is also called an embedding, which is safer to use than immersion, because it is closer to the terminology used in differential geometry.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.