Is There a notion of Convergence for the Non-Standard Reals? How can we do Analysis?

Question

I was reading up on Non-Standard Analysis from an Analytical perspective, i.e., how to do Analysis when I got stuck thinking of how we can do Analysis without even having a "traditional" metric , e.g., if $x$ is a(n) non-real infinitesimal then $d(x,0)=x$ is non-Real, which is not allowed in traditional metrics. But then if the topology is not metrizable, just how is it given? I am curious because I am having trouble thinking of notions of convergence. How about " Physical/Geometrical" models for the Non-Standard Reals in the sense of the Real line being a model for the axioms for the standard Reals? Sorry, I am trying to clarify my thoughts here (Clearly without much success so far ). In my basic research I found that the minimal conditions to be able to do Analysis in a Topological space X is that X must be a Topological Field where 0 is a limit point of $X-\{0\}$. Is this correct?

Answer 1

Not sure whether this is the kind of answer that you wanted or expected.

You only need a topology to talk about convergence in general. To get a nice and convenient topology on non-standard sets ${^*}Y$, one usually takes superstructures (actually in a way such that ${^*}Y$ is an enlargement) and defines any union of sets from ${^*}\tau$ as open, where $\tau$ is a topology on the set $Y$.

What you do in non-standard analysis (instead of non-standard topology) is rather to reformulate analytical notions like convergence in terms of other notions. For example we have that $$\lim_{n \rightarrow \infty} a_n = b$$ if and only if $\text{st}(a_N) = b$ for all $N \in {^*}\mathbb{N}\setminus \mathbb{N}$, where "$\text{st}$" denotes taking the standard part.

Answer 2

Just look at the usual $\epsilon$-$\delta$ definition of convergence:

$\lim_{x\rightarrow a}f(x)=L$ iff for every $\epsilon>0$ there is a $\delta>0$ such that for all $x$, if $0<\vert x-a\vert<\delta$ then $\vert f(x)-L\vert<\epsilon$.

This makes perfect sense in the hyperreals, without changing anything: just make sure that all the variables are allowed to range over the hyperreals. So, for example, to show that $\lim_{x\rightarrow c}x=c$, we just set $\delta=\epsilon$ as usual.

Note that this definition applies to any function ${}^*\mathbb{R}\rightarrow{}^*\mathbb{R}$. Of course, we're usually really interested in the ones which come from functions on $\mathbb{R}$; given such an $f$, in the nonstandard setting we replace $f$ with ${}^*f$ and go as above. We can then use the transfer property to show that everything we're going to get in this context is actually true in standard analysis too.

Similarly, to tell whether a sequence $X=(x_n)_{n\in\mathbb{N}}$ from the standard world converges to some standard real $L$, we first pass to its nonstandard version ${}^*X=({}^*x_n)_{n\in{}^*\mathbb{N}}$ and then ask, in the hyperreal world, the usual question: is it the case that for all $\epsilon>0$ there is some $n$ such that for all $m>n$ we have $\vert {}^*x_m-L\vert<\epsilon$? Transfer tells us (as usual) that this gives the desired result.