UNCLE!
I was going to say that I saw a horsie and a doggie, but I think I’ll just shut up instead. 
[sub]I got to basic differential calculus in college. How much education would I have to get to even start understanding the subject of this thread?[/sub]
If you got through differential calculus and you haven’t completely forgotten it, then you know more or less enough to basically understand the question. To be able to answer it, you need a little more.
The hyperreals are made of sequences of real numbers. Two sequences represent the same hyperreal if the set of the terms where they disagree is not large. What exactly it means to be a large set would take a little bit more explanation, but it’s pretty clearly defined. So you take the hyperreals represented by sequences like (r, r, r, …), and you identify that with r to get the standard hyperreals. But once you’ve done that, you’re left with sequences like (1, 2, 3, 4, …) and (1, 1/2, 1/3, 1/4, …). Those are the non-standard hyperreals. The first is bigger than every standard hyperreal number, and the second is smaller than every standard hyperreal number except 0. And then you can add, subtract, multiply, and divide the hyperreals.
If you remember the notion of intervals from differential calculus, you might remember open intervals (those that don’t contain their endpoints), closed intervals (those that do), and intervals neither open nor closed (cause they contain one endpoint but not the other). Topology is really the study of a generalized notion of open intervals called open sets, and a set of points is completely characterized by its open sets (at least as far as topologists are concerned).
So the question in the OP is this: If you have two different notions of which sets are large, do you get different open sets in the hyperreals? And if not, what do the open sets look like?
Does that make any sense?
Point-set topology, which by now is practically equivalent to analysis. Following (and significantly extending) Klein, any geometry is the study of properties preserved under a certain category of maps. Open sets are actually a bit of a hack, since continuous functions reflect open sets. What’s preserved by continuity is the notion of limits (in general of nets). When properly examined, this can be stated without ever talking about points or sets at all.
As a response to the question posed, though, spot on. Bravo.
Sunspace writes:
> I was going to say that I saw a horsie and a doggie, but I think I’ll just shut up
> instead.
Then you’re in luck. In the topology of the hyperreals, the horsie and duckie are equivalent.