An infinite union of closed sets?

[Castilla]

Er, Hurkyl, maybe you can tell me if that was Ok?

 

[HallsofIvy]

I have just realized something 'illogical' about the definition of open sets according to 'R contains none of its boundary points'. Surely 'its boundary points' refer to R's boundary points so R has boundary points. But than to state R contains none of its boundary points is confusing.

You do understand, I hope, that "A is closed if it contains all of its boundary points" is not intended to be a strict, logical definition. It is shorthand for "If the statement (if x is a boundary point of A the x is contained in A) is true then A is closed". For A= R, the statement "if x is a boundary point of R then x is contained in R" is true for all R because the premises are false.

If every statement had to be in rigorously true logic, then every mathematics book would be as long (and as hard to read) as "Principia Mathematica"!