MHB Fixed Point Theorem & Contractive Maps

ozkan12
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Please give an example of contractive map which have fixed point...I search but I didnt find
 
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Hi,

A trivial example is the constant real function 1.
 
Another trivial example is the map $x \mapsto kx$ for $0 \leq k < 1$ over, say, the reals. It has an obvious fixed point: zero.
 

Thread '2-sphere intrinsic definition by gluing disks' boundaries'

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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