Undergrad Using the Baire Category theorem

[Mr Davis 97]

In my textbook there is the following paragraph:

"The usual application of the Baire category theorem is to show that a point $x$ of a complete metric space exists with a particular property $P$. A typical argument runs as follows. Let $X = \{x\in M \mid x \text{ does not have property } P\}$. By some argument, we show that $X$ is of first category. Since $M$ is of second category (by the Baire category theorem), there exists $x\in M \cap (M\setminus X)$. Thus there exists an $x$ with property $P$."

Could someone explain this a little bit? In particular, Why does $X$ being of first category and $M$ being of second category imply that there exists $x\in M \cap (M\setminus X)$?

 

[martinbn]

Could someone explain this a little bit? In particular, Why does $X$ being of first category and $M$ being of second category imply that there exists $x\in M \cap (M\setminus X)$?

That simply means that $X$ is not all of $M$. If it were, then $M$ wouldn't be of second category, right? So there are point in $M$, that are not in $X$.

 

[wrobel]

For example it is not hard to show that the set of functions those are differentiable at least at a single point has the first Bair category in $C[0,1]$. Thus there are a lot of continuous but nowhere differentiable functions