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Definitions: Let S be a subset of R^n. S is called "open" if it contains none of its boundary points and S is "closed" if it contains all of its boundary points.

1) Let S={(x,y,z) E R^3 | z=0}.

1a) What is the boundary of S?

1b)Is S open, closed, both, or neither?

My attempts:

1a) The boundary of S is S itself, am I correct?

1b) S is closed since every point in the given plane S is a boundary point and S certainly contains every point in the given plane S, i.e. S contains ALL of its boundary points. Is this correct?

Now do I have to check separately that S is "not open" (how?), or can I conclude immediately that "S is closed implies S is not open"?

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2) "R^n and the null set are BOTH closed and open." I have no clue why this statement is true. How can a set be BOTH closed and open? I am just so lost...

Thanks for helping!