Nature of the Continuum When we construct the vector space $\mathbb{R}^n$ we state that every element $x$ is a limit of some rational Cauchy sequence $\{x_i\}$. Two Cauchy sequences $\{a_i\}$ and $\{b_i\}$ are equal if they are eventually $\epsilon$-close together, when $i,j\in  \mathbb{N}$ are big enough, for every small rational number $\epsilon$.
Does this mean that every rational point $x$ in the space $\mathbb{R}^n$ is actually an "extended point" $B_\delta (x)$, where $\delta$ is some ultra small rational number? If this is the case then due to overlapping of those open balls it's easy to understand why $\mathbb{R}^n$ is connected and why continuity exist.
 A: I'm not sure this addresses your question - I don't know what you mean by "extended point," "ultra small," or your last sentence - but this might be helpful:
There is a sense in which each real number is "thick" - a real number is not a Cauchy sequence, but rather an equivalence class of Cauchy sequences. We can think of individual Cauchy sequences as being "ultra close" to each other - for instance, $(0, 0, 0, 0,  . . .)$ and $(1, 0.1, 0.001, 0.0001, . . .)$ are "ultra close" - but the real numbers are formed by "smooshing together" all the ultra-close Cauchy sequences. That is, we have an equivalence relation on Cauchy sequences (two Cauchy sequences are equivalent if the differences of their terms go to zero), and a real number is just an equivalence class of Cauchy sequences.
Now this does mean something slightly surprising: a rational number is not the same as a rational number! Okay, what on earth do I mean by that? Well, the rational number ${1\over 2}$ (e.g.) is not a priori the same thing as the equivalence class of Cauchy sequences which includes $({1\over 2}, {1\over 2}, {1\over 2}, . . . )$ - the latter is a set of sequences of rational numbers, and the former is just one single rational number. We (after the fact) identify the two. So there is something a bit surprising going on, but it's really a type issue, not something about open balls.
A: No, every rational point in $\mathbb{R}$ is just a point. Take a Cauchy sequence $\{a_i\}$ which represents $x$, and a Cauchy sequence $\{b_i\}$ which represents some point you think is in the "extended point" $B_\delta(x)$, but which is NOT equivalent to the Cauchy sequence $\{a_i\}$. Since the two Cauchy sequences are not equivalent, that means that for a certain $\delta$ and a certain $N$, for all $i,j>N$, $|a_i-b_j|>\delta$. Thus, there is no sense in which $x$ contains Cauchy sequences that are "ultra" close to it, but not equal to it.
