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I am taking a math class where the book used is Walter Rudin. I don't understand how the author explain that in $R^2$ the complex number such $|z|<1$ is

  1. not closed

  2. open

  3. not perfect

  4. bounded

My other question is: how does the distance function change the way we define the neighborhood on the set, and for subset? How do we apply the definitons of: open, closed, complement, perfect, dense... on the topological space $(X,d)$ with a particular $d$ or a new definition of $d$ that is different of the usual distance we operate with?

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