In the same vein as in theorem $2.33$ in Rudin's PMA book (where he proves that compactness is absolute), I am trying to prove that connectedness is also an absolute notion.
That is to say that if $Y\subseteq X$ and $E\subseteq Y$ then $E$ is connected relatively to $Y$ if and only if it is connected relatively to $X$.
So I started writing the definitions (trying to prove $E$ not connected in $X$, if and only if, $E$ not connected in $Y$) :
E is not connected in $Y$ if $\exists A$ ,$B$ such as:
- $A$, $B$ are open in Y
- $A$, $B$ $\ne \emptyset$
- $A\cap B = \emptyset$
- $A\cup B = E$
and:
E is not connected in $X$ if $\exists A$ ,$B$ such as:
- $A$, $B$ are open in X
- $A$, $B$ $\ne \emptyset$
- $A\cap B = \emptyset$
- $A\cup B = E$
. However I've got a problem with the second definition: how can $E$ be equal to $A\cup B$ with $A$ and $B$ open in $X$ ? If $Y=\mathbb R$ and $X=\mathbb R^2$ then, for an open set $V$ of $X$, we must have necessarily $V \cap Y^c \ne \emptyset$. But then, $E=A\cup B$ (with $A$ open in $X$ and $B$ open in $X$) would imply that $E\nsubseteq Y$. Which contradicts the assumptions !