Show that both [itex]Y\cup A[/itex] and [itex]Y\cup B[/itex]

[Maths Lover]

Homework Statement

Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected.

The Attempt at a Solution

I can show easily that at least one of $Y\cup A$ or $Y\cup B$ is connected. but I don't know how to show the required

Any hints?

Here are some facts that I know about connectdness that may be helpful.

As a way to proceed, we can suppose that There is a separation $C \cup D$ of,say, $Y\cup A$ So it's clear that $Y$ included in either $C$ or $D$

 

[haruspex]

I'd say that's good start, and you can choose C to be the one containing Y.
Maybe you can now show that D, BUC form a separation of X? Or something like that.

 

[andrewkirk]

As a way to proceed, we can suppose that There is a separation $C \cup D$ of,say, $Y\cup A$ So it's clear that $Y$ included in either $C$ or $D$

Yes that is how I would start. Let's call C the part that includes Y.
Next we can use the lemma (1.1 on page 47 of my very old edition of 'Topology - a first course' by Munkres) that if two sets form a separation then neither contains any limit points of the other, So C contains no limit points of D, and also B contains no limit points of D because $D\subset A$ and A,B is a separation of X-Y.

But then $D, C\cup B$ is a separation of $X$ because
$\{x\vert x$ is a limit point of $C\cup B\}\subset \{x\vert x$ is a limit point of $C\}\cup\{x\vert x$ is a limit point of $B\}\subset X-D$

Since $X$ is connected we conclude that there can be no separation $C,D$.