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, is it 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 logic? 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!Please Register or Log in to view the hidden image!
Yes. Basically, yeah, although it's probably sufficient to just note that every point in S is a boundary point. No, there are sets that are both open and closed, so you can't immediately make that conclusion. So I'd do it directly in this case, which is easy. I.e., if a set is open, every point in that set should have a neighborhood that is also in the set. Since any movement along the z axis will take you out of the set, you can see that the neighborhood of any point in S is not in S, and so S is not open. These are both sort of pathological cases. Note that the null set and R^n are complements of one another, so if one of them is both closed and open, it follows that the other must also be both closed and open. First consider the null set. The closure of the null set is again the null set, so you can immediately see that it is its own closure, and so is a closed set. Similarly, the condition that the neighborhood of any point in the set also being in the set is trivially true, since there are no point in the set. So it's both closed and open. You can proceed in a similar way with the R^n.
Is 'boundary' defined rigorously here? If not, this definition is a bit fluffy. If you have any trouble understanding quadraphonics' post, you'll probably need to review the definitions. Usually a set is defined as open if every point in the set has a neighbourhood also in the set. Closed sets are just complements of open sets.
Just follow the definition. The meaning of some words in mathematics are different than their everyday use. It is normal for a set to be both open and closed.
