What is the Definition of C1-Close Curve in Whitney Topology?

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SUMMARY

The term 'C1-close' refers to the concept of curves being close in the C^1 topology, specifically in the context of Whitney topology. Two curves, c1 and c2, are defined as C^1-close if there exists an epsilon > 0 such that the conditions |c1(t) - c2(t)| < epsilon and |dc1/dt - dc2/dt| < epsilon hold for all t in the interval [0,1]. This definition is crucial for understanding the behavior of curves in differential topology, as detailed in the reference "Differential Topology" by Morris Hirsch.

PREREQUISITES
  • Understanding of C^n topology
  • Familiarity with Whitney topology
  • Basic knowledge of differential calculus
  • Exposure to differential topology concepts
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  • Study the C^n topology in detail
  • Read "Differential Topology" by Morris Hirsch
  • Explore the implications of Whitney topology in geometric analysis
  • Investigate applications of C^1-close curves in mathematical modeling
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Mathematicians, students of topology, and researchers in differential geometry will benefit from this discussion, particularly those interested in the properties of curves and their applications in advanced mathematical theories.

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I'm reading a paper and have came across the term 'Cn-close' in the sense of a curve being C1-close to a circle for example, but can't find a definition of this term anywhere, and would be grateful if anyone could help.
 
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This is a reference to the C^n topology, or Whitney topology: http://en.wikipedia.org/wiki/Whitney_topologies

In your case, to say that " As soon as two curves c1, c2: [0,1] --> R² are C^1-close together, then "blahblah"" means that there exists epsilon >0 such that whenever |c1(t) - c2(t)| < epsilon and |dc1/dt - dc1/dt| < epsilon for all t, then "blah blah" holds.

A reference is Differential Topology by M Hirsch.
 

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