Recent content by scott

I've read that experiments have been done ( by whom escapes me at the moment) where the phase-states between changes in states of various liquids were studied mathematically, and the mandelbrot set did in fact emerge at certain points in the phase-space. As for the planets, yes they do look...

I've been thinking, and I've come up with a sort of Intelligent Design argument in reverse against the existence of a God. It occurs to me that perfect shapes, such as circles, squares, rectangles, right triangles, etc .. are the product of an intelligent mind. In other words it takes an...

I'm not actually in a course. Just a soul who's insatiably curious about the universe and how it works. My intermediate goal is to understand the mathematics of general relativity. The most formal instruction I've had in math would be up through introductory calculus. I've gone back and reviewed...

My bad. I'll re-read the definitions. ditto Just about every place I've seen defines the set of ordered pairs that's way, that's why. here's a snippet from a mulivariable calculus book: I can find many other examples of the Cartesian product defined in just this way. I was...

I'll double-check my use of those terms again. I meant R^2 and was using that as an example with which to demonstrate the relationship between different types of spaces all using that example. I understand that the Pair you referred to (X,T) is not the ordered pair in my example. Again, I...

Ok thanks for the replies guys. One thing I've been struggling with a little is the idea of moving from the specific to the general in going from definitions of Euclidiean to Metric, to Topological spaces. I think I may be on the right track. Here's a specific example to illustrate going from...

I mistyped above, I meant to say that the set X is the topological space. I don't think I'm that far off, I realize now that X is the topological space which has the set T, a subset of the power set of X, as a topology on it. Forgive my non-rigorous, non-formal language, but I picture a...

The way I understand it, a topological space T, is defined as a subset of a set X, or a topology on X. Is this definition meant to include the entire set X, or only a specific set contained within X, who's elements are less than the total elements contained in X (i.e., a specific "part" of X)...

Here's the http://planetmath.org/encyclopedia/TopologicalSpace.html I'm looking at. Like I said, I'm a noob at this, trying to digest all of this new terminology, maybe I'm making it harder than it is. Here's another definition

Ok, this occurred to me last night about my first question. Back to the definition of a topological space, if it is defined as a collection of subsets T of the set X, and X is a member of the collection of sets, doesn't that define the entire set X?

lethe, yes your answer helped. matt: Is that how we manage to understand four dimensional space-time that we can't really visualize geometrically, but because we know the mathematical rules that apply at lower "visualizable" dimensions, we can simply follow the rules to apply them at...

Ah, yes you are right, my sloppy reading of the definition of a subset. The very next sentence after my prior quote Here's another idea I'm struggling with though. The distinction between a closed set and an open set. If I'm not mistaken, an open set of the real numbers over the...

I'm a noob starting out studying differential geometry and topology. Really probably somewhere in the multivariate calculus level, but I've been trying to understand the plethora of terminology I'm encountering with this higher math. I've been reading a lot on Wikipedia.org and PlanetMath.org...

So are you saying then, that massive objects which exist create a field between them, a gravitational field, and this field can be flat or curved depending on the mass of the objects? Is the gravitational field anything like an electrical field or a magnetic field? Is the gravitational field...

What is space-time "made out of"? A thread in another forum about the definition of "nothing" got me thinking. I know matter is composed of molecules, which are composed of atoms, which are composed of electrons, neutrons and protons, which are composed of quarks, etc ... But what about...