- #1
kingwinner
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Hi, I have some questions regarding open and closed sets.
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!
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"?
=================
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!