- #1
Mr Davis 97
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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)##?
"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)##?