Munkres Lemma 23.1

Question

In the second paragraph, Munkres assumes that there exists a separation of $Y$ (in the sense he defined in Lemma 23.1) and proves that $Y$ is not connected.

So I would think the first paragraph should prove the converse statement: if $Y$ is not connected, then there is a separation. But in the first paragraph Munkres says "Suppose $A$ and $B$ form a separation of $Y$". Is it true that he does NOT mean the separation that he defined in the statement of Lemma 23.1? Does he mean "Suppose $Y$ is not connected"? If so, the next sentence that claims that $A$ and $B$ are open in $Y$ makes sense, but otherwise (if $A$ and $B$ form a separation in the sense he defined in Lemma 23.1) it doesn't.

Answer 1

I agree that this is unclear. It looks to me as though Munkres means “Suppose $Y$ is not connected.”

The first use of “separation” in the proof seems to be his way to say that $A$ and $B$ form a witness to the non-connectedness of $Y$: they partition $Y$ into two disjoint open (in $Y$) sets, and not that they form a separation as defined at the beginning of the lemma.

Whether that’s carelessness or perhaps because he uses the term “separation” to mean something else in his definition of connected, I don’t know. (Additionally suggestive of some carelessness, the statement of the lemma is a one-way implication, but he seems to prove an if-and-only-if proposition, and some writers might argue that beginning a lemma statement with an “If $P$, $Q$” sentence that is intended as a definition, and not the sentence to be proved, is also a problem.)

Answer 2

In my understanding, $Y$ can be viewed as a subset then the first paragraph is the proving of $\overline{A}\cap B=\emptyset$ and $A\cap\overline{B}=\emptyset$. This means $A,B$ is a pair of separated set. In "Counterexample of Topology" this lemma is written as the definition of connected set.