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I have 'learned' the basic definitions of neighborhood, limit point, closed, and closure but have some trouble accepting the following examples.

1. For Q in R, Q is not closed. The set of all limit points of Q is R, so its closure is R.

Between every two real numbers is a rational, I know as fact. But can't it be that between any two rational numbers is another rational?

- Since we can adjust the neighborhood of any rational to be of any radius, won't we cover all rational numbers in this way? So is it incorrect to say that the limit points of Q can be Q itself?

- Or is it simply a matter of being as complete as possible? Not complete in the real analysis sense, but in the sense that possibilities are included. It seems more complete to account for any point that can be a limit point of Q. In this case, I know it's correct to say more generally that R is the set of all possible limit points of Q, since Q is in R.

2. E = { 1/n | n = 1, 2, 3, ... }

The limit point of E in this case is 0, so the closure of E is E U {0}.

This is the set of rationals in (0,1] of the form 1/n. So 1 cannot be a limit point since we can chose the neighborhood of 1 to be, say 0.1, and it will not include any other points of E.

- Is this the same reasoning behind saying that no other 1/n in E can be a limit point? And the same reasoning behind saying that [0,1] in R is not the set of all limit points of E?

- But why then is zero a limit point? Can't we adjust its neighborhood so that it does not cover a number of the form 1/n? Are there no non-reducible rationals that are very close to zero that

*don't*have the form 1/n, some tiny number over a large number?

It seems like the best form of such a number would be of form 1/n, but... Do all rationals near zero have the form 1/n? This is what the example seems to be saying since we can pick any neighborhood, yet 0 is still a limit point.

Any clarity would be appreciated... Pardon my slowness but I want to know what mistakes I'm making here. Thanks :)