/sci/ - Science & Math

>>12002002
The unbounded 1D open interval can be defined in any number of ways. There is a well known theorem in topology which proves that all unbounded 1D open intervals are the same object, and there are various proofs of the same outside of topology. You can define the unbounded 1D open interval as a Hausdorff space, for example, without making reference to R.

>but theres more than one,
You are wrong. There is only one.

>distinguish your R from the normal Real number line and from the long line in topology.
Wrong again. The long line is the closure of the unbounded 1D open interval. Since the unbounded 1D open interval is not clopen, the long line is exempted from the class of objects which are all the same. BTFO.

>But even before then, an interval of What
>you cant define an interval without defining the shit inside it
It's an interval of points. From Wikipedia: "In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other." BTFO.

>inb4 what is a point?
A point is an idea whose existence mathematicians concede: "a primitive notion upon which geometry is built," according to Wikipedia. The triangle ABC is constructed from the points A, B, and C, for example.

>the symbol (-inf, inf) is defined to mean the set of all real numbers
Since circular definitions are not desirable in my work, I have assigned the definition in the other direction. I can avoid circular definitions by saying R is equal to the interval, and then defining the interval as a Hausdorff space without referring in a circular fashion back to R.