What does Liu mean by "topological open/closed immersion" in his book "Algebraic Geometry and Arithmetic Curves"?
Solution 1:
Yes, that's a correct definition. Yours (1.) is also equivalent to 2. below.
- $f(X)$ is open (closed) and $f$ is a homeomorphism on its image
- $f$ 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.