MHB Fixed Point Theorem & Contractive Maps

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A contractive map example provided is the constant real function, which always returns the value 1. Another example is the map defined by $x \mapsto kx$ for values of k between 0 and 1, which has a fixed point at zero. These examples illustrate the concept of fixed points in contractive mappings. The discussion emphasizes the simplicity of these examples in understanding fixed point theorems. Overall, contractive maps can have straightforward fixed points that are easy to identify.
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 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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