I have these two propositions stated in my book as follows.
Proposition 1.
Let be a metric space, then
a) The sets and are open
b) Any finite intersection of open sets is open
c) Any union of open sets is open.
Similarly,
Proposition 2.
Let be a metric space, then
a) The sets and are closed.
b) Any finite union of closed sets is closed
c) Any intersection of closed sets is closed.
I understand parts b) and c) of both propositions. However, I have no clue why a) is listed for both and why a) is important. What is a) telling us? If someone could please explain to me the meaning of a) in both cases it would be much appreciate. Thank you very much.
Definition of "open set"- A is open if and only if, "If x is in A, then there exist d> 0 such that N(x,d), the set of all points whose distance from x is less than d, is a subset of A.
If A is itself the set of all points, what does that tell you? If A is empty what happens to the hypothesis "if x is in A"?