My book gives a very tantalizing yet brief note that the Baire Category Theorem can be used to show that a point $x$ in a complete metric space $M$ has a particular property $P$. It states:
"A typical argument runs as follows. Let $X = \{x \in M : x \space \space \text{does not have property}\space \space P\}$. By some argument, we show that $X$ is of first category. Since $M$ is of second category, there exists $x \in M \cap X'$. Thus $x$ has property P."
My confusion here is why there exists an $x \in M \cap X'$ and also if the above is a proof by contradiction (since it seemingly states $x$ both does and doesn't have property $P$.)
Could anyone give me some insight to this footnote in my book? Thank you!
It can be re-worded as proof by contradiction, if you like. Proof (by contradiction): Assume no element $x\in M$ has property $P$. Then $X$ would be equal to $M$. If by some argument you had that $X$ is of first category, but $M$ is of second category, and since (by definition) no space can be of first and second category at the same time, you obtain a contradiction.
Second proof (without contradiction). Let $M$ be of second category and suppose by some argument you showed $X$ is of first category. Since it is then impossible that $X=M$, and since $X\subseteq M$ by definition, we conclude that $X\subset M$. That means, that there is an element $x\in M$ which is not in $X$.