12:42 AM
So, I was reading about IST, an extension of ZFC...
In IST, $\Bbb R$ has "nonstandard" elements, such as infinitesimals and infinitely large numbers.
An infinitesimal is one with a magnitude smaller than any standard real; an infinitely large number (an "unlimited number") is one larger than any standard real.
Two numbers $a$ and $b$ are "infinitesimally close to each other" if $a-b$ is infinitesimal.
And every limited (i.e. not unlimited) real is infinitesimally close to some standard real number. The one infinitesimally close to $a$ is denoted $\operatorname{st}a$.
Cool thing is: Definition of compact spaces are so much simpler!
A space $A$ is compact iff, for all $a\in A$, we have $\operatorname{st}a\in A$.
For example, $(0,1)$ isn't compact: Let $\epsilon$ be a positive infinitesimal. It's in $(0,1)$ because it's between $0$ and $1$. However, $\operatorname{st}\epsilon=0\not\in(0,1)$.
Similarly, $[0,\infty)$ isn't compact: Let $N$ be an unlimited number. Then $\operatorname{st}N$ doesn't exist, so $\operatorname{st}N\not\in[0,\infty)$.
(If you think this is too specific to subspaces of $\Bbb R$: Don't worry, this all extends to arbitrary spaces.)
(If $X$ is a standard space, then (for $t$ standard) $\tau$ is infinitely close to $t$ iff $\tau$ is in every standard neighborhood of $t$.)
(I forgot to mention: The above definitions all assume that the spaces are standard. However, there's an axiom of IST that basically lets us extend definitions like this to all spaces.)
(Also, I think — but have not verified — that we get a nonstandard version of Hausdorff spaces, too: I think $X$ is Hausdorff iff, for $a,b\in X$, we have "$a$ is infinitely close to $b$ iff $b$ is infinitely close to $a$." That is, "infinitely close" is a reflexive relation.)
(Amendment: Don't think $t$ needs to be standard above.)
