Algebraic Geometry: Investigating x_1 and x_2 Generating Structure Sheaf

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Dear all,
Studying for prelims, I would like to know the following. I attached the question as well(sorry).
Why do x_1 and x_2 not generate the structure sheaf on the Affine 2-space/ Thank you for your help.
 

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Thread 'Injective immersion and embedding'

Hello! There is a simple line in the textbook. If ##S## is a manifold, an injectively immersed submanifold ##M## of ##S## is embedded if and only if ##M## is locally closed in ##S##. Recall the definition. M is locally closed if for each point ##x\in M## there open ##U\subset S## such that ##M\cap U## is closed in ##U##. Embedding to injective immesion is simple. The opposite direction is hard. Suppose I have ##N## as source manifold and ##f:N\rightarrow S## is the injective...
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