- #1
rad0786
- 188
- 0
Hi.
Can somebody please check my work!?
I'm just not sure about 2 things, and if they are wrong, all my work is wrong.
1. Find a counter example for "If S is closed, then cl (int S) = S
I chose S = {2}. I am not sure if S = {2} is an closed set? I think it is becasue S ={2} does not have an interior point, and 2 has to be a boundary point, (and 2 cannot be both and interior and a boundary point.) I am sure S={2} is closed!
cl (int S)
=cl (int 2)
=cl (empty)
= empty
and that is not equal to S = {2}
2. Let A be a nonempty open subset of R and let Q be the set of rationals. Prove that (A n Q) ... (I hope those symbols show, I got them from MS Word)
I figured that Since "A is a nonempty open subset of R," A has to be composed of MORE than 1 element, hence, A has to have 2,3,4... elements.
And I know (from lectures and the textbook) that between any 2 real numbers, their is a rational. Hence, (A n Q) .
How does that sound? Can somebody please check this? Thanks in advance.
Can somebody please check my work!?
I'm just not sure about 2 things, and if they are wrong, all my work is wrong.
1. Find a counter example for "If S is closed, then cl (int S) = S
I chose S = {2}. I am not sure if S = {2} is an closed set? I think it is becasue S ={2} does not have an interior point, and 2 has to be a boundary point, (and 2 cannot be both and interior and a boundary point.) I am sure S={2} is closed!
cl (int S)
=cl (int 2)
=cl (empty)
= empty
and that is not equal to S = {2}
2. Let A be a nonempty open subset of R and let Q be the set of rationals. Prove that (A n Q) ... (I hope those symbols show, I got them from MS Word)
I figured that Since "A is a nonempty open subset of R," A has to be composed of MORE than 1 element, hence, A has to have 2,3,4... elements.
And I know (from lectures and the textbook) that between any 2 real numbers, their is a rational. Hence, (A n Q) .
How does that sound? Can somebody please check this? Thanks in advance.