How was born the topology from Euclidean geometry? Good evening, I have the question arose as to create interest starting topology of Euclidean geometry, what was the interest of those who created it?
Thanks for your help

You have to distinguish between general topology and algebraic topology.  For general topology, a key idea was to define "closeness" for Riemann surfaces, in order to give complex functions such as  $\surd z, \log z$ a single value by defining them on something lying "over" part of the usual complex plane $\mathbb C$. This led to Hausdorff's notion of "neighbourhood space" and then to the logically simpler, but less intuitive,  notion of a topology defined by open sets. 
The origin of algebraic topology lies first with Euler's solution of the Konigsberg Bridge problem, and then later with workers defining Betti numbers and torsion coefficients of higher dimensional spaces. This led to the problem of defining notions such as "boundary" and "cycles" so that one could think of Betti numbers as arising from "cycles modulo boundaries". Poincare in his great series of papers on "Analysis Situs", later called "Topology", solved this problem by taking the free abelian group on the oriented simplices of a given dimension $n$ and defining the boundary $\partial$ of an $n$-simplex $\sigma$ by 
$$\partial \sigma= \sum_\rho \epsilon(\sigma,\rho] \rho  $$
for $\rho$ of dimension $n-1$ in the boundary of $\sigma$, and $\epsilon$ denotes the chosen orientations. This was much later simplified by Eilenberg by choosing an ordering of all the vertices so that one could define 
$$ \partial \sigma = \sum _i (-1)^i \partial _i \sigma $$
leading to the crucial rule $\partial \partial =0$, so that every boundary $z= \partial \sigma$ was a cycle, i.e. $\partial z = 0$. 
