Why will we make an error if we are using the concept of Euclidean neighborhood and metric on the hyperbolic plane?

In the literature the derivation of the hyperbolic numbers is often introduced by the Euclidean neighborhood and metric on the hyperbolic plane. I present the related problems based on an article¹ on the hyperbolic calculus.

Quote from the above mentioned article of A. E. Motter and M. A. F. Rosa:

“Now we turn to the hyperbolic numbers, P= M². We employ in M² the topological structure of R², this could be seen as contradictory and is criticized (see [9]²). But it is the convention adopted in all the literature (see [10,11] for example) and we assume here this simplified point of view (leaving the suggestion made in [9] for a posterior work). This means that, despite of the Lorentzian structure of M² we will be using the concept of neighborhood and making limits as if we were in R².”³

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¹See http://www.researchgate.net/publication/225328790_Hyperbolic_Calculus

² See Zeeman E. C., The Topology of Minkowski Space, Topology, 6 (1967) 161.

³ Notations: P means the hyperbolic plane, M² indicates the Minkowski 2-dimensional space time, R² signifies the plane with the Euclidean metric.

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