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My textbook has the following statement in one of its chapters:

Chapter 8:Topology of R^n

*If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now.*

Chapter 10 covers topological ideas in a metric space setting. I understand what a metric space is (a set of points over which there is defined a distance function that satisfy three special properties), but I don't understand how the three concepts mentioned in the title are related.

1. Are every euclidean space a metric space? is the converse true? are all metric spaces a euclidean space? why not?

2. What is the meaning of "topology of Euclidean spaces"? How is topology related to metric spaces?

I appreciate you any input you guys can contribute. Thanks.

M