Is this statement about connected sets true? Statement: $A\subset B \subset X$ with subset $B$ equipped with subspace topology then "$A$ is connected in $B$ if and only if $A$ is connected in $X$" 
I am not able to find counter example of above statement! Is it true? I know that if word connected is replaced by compact then it is true!! But, for connected is it true? Please help me..
 A: It holds as a consequence of the fact that the topology $A$ inherits as a subspace of $B$ is the same as the topology $A$ inherits as a subspace of $X$.
Namely, if $U \subseteq A$ is open relatively to $B$, then there exists $V \subseteq B$ such that $U = A \cap V$. $V$ is open in $B$ so there exists $W \subseteq X$ open in $X$ such that $V = W \cap B$.
We have $$U = A \cap V = A \cap (W \cap B) = A \cap W$$
so $U$ is open relatively to $X$. The converse is similar.
A: Yes, it is true. That's because the subspace topology induced on $A$ by the topology from $B$ is the same as the subspace topology on $A$ induced by the topology on $X$. And this is trua because, if $A^*\subset A$, then asserting that $A^*$ is open with respect to the subspace topology induced on $A$ by the topology from $B$ means that $A^*=B\cap Z$ where $Z$ is an open subset of $B$. And asserting that $A^*$ is open with respect to the subspace topology induced on $A$ by the topology from $X$ means that $A^*=B\cap W$ where $W$ is an open subset of $X$. But these assertions are equivalent: in order to pass from the second one to the first one, just take $Z=B\cap W$. And in order to pass from the first one to the second one, take an open subset $W$ of $X$ such that $Z=B\cap W.$
