Help finishing connectedness proof.

[Mister Eric]

Been having trouble with connectedness. Tried out a problem to practice and not sure how to finish it.

The problem is this:Let A and B be closed subsets of the space X. If A U B and A n B are connected, so also are A and B. Show by counter-example that both A and B must be closed, i.e., if one of A and B is not closed then in general the thm fails.My proof went like this:

Let C = A ᴜ B. Suppose that U, V give a separation of C, where U and V are open in C.
Since A is connected, we must have A ᴜ U or A ᴜ V (otherwise, A ∩ U and A ∩ V give a separation of A).
Without loss of generality, assume A ᴜ U. Since A ∩ B = Ø and B is connected,
we must have B ᴜ U. This implies that C ᴜ U, and therefore, V = Ø.
This contradicts that U, V is a separation of C. Hence the sets A or B must be closed,The response I got back was this:

To prove your result, assume (to get a contradiction) that one of the closed sets, A or B is not connected. WOLOG, we may assume it is A. Then there are open sets V and W in X s.t. A= (A ∩V) U (A∩ W) and = (A ∩V) ∩ (A∩ W)=ᴓ.
Now, you should be able to use V and W to build a separation of either AUB or A∩B. (Try “unioning“ B to both sides of the above equations, and it may give you an idea of what to use for a separation.)I thought I was doing it right, but by my professor's response, I'm not sure. Could anyone tell me what I'm doing wrong and what he means by "unioning" B to both sides?

Any feedback would be greatly appreciated.

 

[DonAntonio]

Been having trouble with connectedness. Tried out a problem to practice and not sure how to finish it.

The problem is this:


Let A and B be closed subsets of the space X. If A U B and A n B are connected, so also are A and B. Show by counter-example that both A and B must be closed, i.e., if one of A and B is not closed then in general the thm fails.


My proof went like this:

Let C = A ᴜ B. Suppose that U, V give a separation of C, where U and V are open in C.
Since A is connected, we must have A ᴜ U or A ᴜ V (otherwise, A ∩ U and A ∩ V give a separation of A).
Without loss of generality, assume A ᴜ U.

You already repeated this several times to be a mere mistake: you write "we must have \,A\cup U\,\,or\,\,A\cup V\,", and then again " assume \,A\cup U\,"

WHAT are we assuming, anyway?? My guess is that you actually mean "assume \,A\cup U=A\," , but then what do you expect from this?

I think you first must be clear above in order to meet your instructor's criticism.

Since A ∩ B = Ø and B is connected,
we must have B ᴜ U. This implies that C ᴜ U, and therefore, V = Ø.
This contradicts that U, V is a separation of C. Hence the sets A or B must be closed,


The response I got back was this:

To prove your result, assume (to get a contradiction) that one of the closed sets, A or B is not connected. WOLOG, we may assume it is A. Then there are open sets V and W in X s.t. A= (A ∩V) U (A∩ W) and = (A ∩V) ∩ (A∩ W)=ᴓ.
Now, you should be able to use V and W to build a separation of either AUB or A∩B. (Try “unioning“ B to both sides of the above equations, and it may give you an idea of what to use for a separation.)


I thought I was doing it right, but by my professor's response, I'm not sure. Could anyone tell me what I'm doing wrong and what he means by "unioning" B to both sides?

Any feedback would be greatly appreciated.

DonAntonio