Examples
If and have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf and corresponding germs: some notable examples follow.
- If are both topological spaces, the subset
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- of continuous functions defines germs of continuous functions.
- If both and admit a differentiable structure, the subset
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- of -times continuously differentiable functions, the subset
- of smooth functions and the subset
- of analytic functions can be defined ( here is the ordinal for infinity; this is an abuse of notation, by analogy with and ∞), and then spaces of germs of (finitely) differentiable, smooth, analytic functions can be constructed.
- If have a complex structure (for instance, are subsets of complex vector spaces), holomorphic functions between them can be defined, and therefore spaces of germs of holomorphic functions can be constructed.
- If have an algebraic structure, then regular (and rational) functions between them can be defined, and germs of regular functions (and likewise rational) can be defined.
Read more about this topic: Germ (mathematics)
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