Been having trouble with connectedness. Tried out a problem to practice and not sure how to finish it.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Help finishing connectedness proof.

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