Which is the difference between a topological space and a euclidean space? Is the topological space a generalization of a euclidean space?

Question

My question arises from a paper in the AMM (Rauch, 1978) in which the illumination problem is treated. The problem is given in an euclidean space, however Rauch uses terms that I think are from the topological figures —$\partial S$: Boundary?, $\bar{S}$: Closure?, etc— I don't sure about the definitions.

Sorry for the English mistakes Thank you!

Answer 1

Euclidean $n$-space is (most commonly) the space $\Bbb{R}^n$, with the dot product giving an inner product. This makes Euclidean $n$-space into an inner product space, which means it is a normed vector space, hence has a topological structure. In summary, a topological space is a huge generalisation of a Euclidean space, and all the notions from topology apply.