I am taking a first (graduate level) course in algebraic geometry and I am trying to understand in what sense the Segre embedding is an embedding, but I am having trouble finding the definition of an embedding in this context. I haven't learned about schemes yet, so in particular, what is the definition of an embedding of varieties?
From the definition of embedding in other contexts, I assume that an embedding algebraic geometry should be injective, (probably) regular and a homeomorphism onto its image. Is this correct or are there other conditions?
Edit: Similar to this question: Segre embedding, though I would like a precise statement of the definition of embedding.
I would define an embedding in this case as a morphism of varieties which is an isomorphism onto its image, and this image can be written as the intersection of an open and a closed subset (of the target domain). This should agree in reasonable cases with what is also referred to as an immersion.