Closure of rational and real numbers
i was wondering what the closure of these both sets of numbers are?
my guess is, that the closure of the rationals are the real numbers, but i have often seen the notation $$\bar{\mathbb{R}} =\mathbb{R} \cup \{-\infty, \infty\}.$$ i don't think that this is a meaningful notation, as this would require a topological space that contains the real numbers as a subset and the elements + and - infinity.
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$\begingroup$Closure depends on the ambient space.
In the real numbers, the closure of the rational numbers is the real numbers themselves.
However often we add two points to the real numbers in order to talk about convergence of unbounded sequences. The reason is that $\Bbb R$ is homemorphic to $(-1,1)$ and the closure of $(-1,1)$ is $[-1,1]$. In this case $\pm\infty$ takes the role of $\pm 1$.
$\endgroup$ $\begingroup$It doesn't make sense to ask if a set is closed: you have to ask whether it's closed in something else!
The rational numbers are closed in the rational numbers. But the closure of the rational numbers in the real numbers is indeed the real numbers.
The real numbers are closed in the real numbers. But the closure of the real numbers in the extended real numbers is all of the extended real numbers.
There are other notions one can talk about, like various sorts of completion or compactification. The extended real numbers are a compactification of the real numbers.
One thing you can do is determine, intrinsically, that the real numbers have two "ends" (I don't recall the precise details of what makes an end an end: it has something to do with compact subsets and their complements); so if you're looking at things this way, compactifying the reals by adding one point for each end is a natural thing to do.
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