let P and Q be two non empty disjoint subset of R . which of the following is /are True ?
a) If $P $and $Q$ are compact , then $P \cup Q $is also compact .
b) If $P$ and $Q$ are not connected,then $P \cup Q$ is also not connected .
I thinks both a) and b) are true...for a ) i take P= [0,1] and Q = [2,3]
for option b) i take P ={x} and Q ={y}...where {x} and {y} are singleton sets...
is my answer is coorect or not ?? pliz tell me
On
A is correct. Your proof is wrong because it requires looking
at all compact subsets of R. You only looked at two.
B is false for all disconnected subsets of R even though there
are many examples of disconnected sets whose union is not
connected. BTW, {x} and {y} are connected.
Q, the rationals, and P, the irrational are disconnected.
Is the union of P and Q disconnected?
b) is false. Let $A=(0,2)\cup \{2.5\}\setminus \{1\}$, $Q=[2,4)\cup \{1\}\setminus \{2.5\}$. Then $P \cup Q=(0,4)$ which is connected though $P$ and $Q$ are not.