MHB Recifiable Paths in C .... Conway, Section 1, Ch. 4 ....

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I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ...

I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ...

I need help in fully understanding some notes by Conway on rectifiable paths in $$\mathbb{C}$$ on page 63 ... ...

The notes on rectifiable paths on page 63 read as follows:https://www.physicsforums.com/attachments/7446
My question regarding the above text from Conway is as follows:Why exactly does $$\phi$$ need to be non-decreasing in order for $$\gamma \circle \phi$$ to be a path with the same trace as $$\gamma$$ ... ... ?Help will be much appreciated ... ...

Peter
 
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He's not saying that $\varphi$ must be nondecreasing, but that if $\varphi$ is nondecreasing, then $\gamma\circ \varphi$ has the same trace as $\gamma$. It's a sufficient condition he's giving, not a necessary condition.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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