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

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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! :)

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1 Answer 1

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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.

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