I'm trying to work out a way to Express the discontinous changes which occur when you take the limit (to infinity) of (for example) the radius of curvature of a spherically closed space - (which appears to approach a flat euclidean space of zero curvature) - ideally i'd like to be able to derive rules for these kinds of transitions which can be generalised for different topological changes... Like the limit of radius of curvature -> 0 for a spherical closed, isotropic space. (or some of the more interesting toroidal cases)

Of course, if this has been done already, i'd like to be able to read the language used... Which i tend to fnd rather painful.

I see. I don't think I quite follow, I fear. The discontinuous changes in what?

I think it is a discontinous change in topology, unless i'm using that word incorrectly... No actually it might be possible for the sphere -> flat change to be continuous... Or maybe i'm ignoring the antipode, yeah, it has to rip at infinity and no longer loop around... And the ripping of space is a discontinous change of topology.

I see.

Well, I think I would say something like, "as its radius becomes arbitrarily large, a sphere can become arbitrarily close to flat [i.e., zero curvature], but can never get there" (due to the Gauss-Bonnet theorem). It's hard to put this precisely in terms of limits and topological changes -- limits deal with what happens as you get close to, but don't reach some limiting point, whereas a "tear" has to happen at some finite point of a process, and is by nature (as you say) discontinuous. Three statements that would be true:

1) In the limit as the radius of a sphere grows without bound, its local curvature goes to 0.

2) In the limit as the radius of a sphere grows without bound, its topology remains exactly the same.

3) There is no topological sphere that has zero curvature everywhere. (Gauss-Bonnet).

So if you want, the space of spheres is not "complete" in this sense. (This is all very informal). Really, though, I would use the language of differential geometry and just say what precisely happens locally and globally, which are often different but subtly related questions. Being too informal with some of these questions can land one in hot water.

Non-euclidean geometry? Try the Necronomicon. It can shew you a lot about that eldritch topic.

The local curvature goes to zero - and 'local' gets bigger and bigger - thus as you get infitesimally close to infinite radius the curvature on any scale goes to zero (for any definition of 'local' ?)

At infinity the quality of 'spherical'-ness (ie: that the space forms a closed loop, with a point which is the maximun distance away, this being the point where turning back makes no sense because it is a shorter distance to keep going forward, based on the standard metric; lets call this the anti-pode) changes - the distance to the anti-pode approaches infinite and thus it approaches not being reachable - this fundamentally changes the quality of the topology.

@smeck - so you're saying i'd be better off using the language of differential geometry as the tool to describe these kinds of mental contortions?

Orathaic,

Right. I think differential geometry is your ticket.

Two things though. I disagree that "at infinity, the quality of spherical-ness changes." As spheres get bigger and bigger, they don't get even vaguely less spherical -- they are all topologically completely equivalent. So I'm not sure I see it as particularly fruitful to say that, topologically, a plane is a limit of spheres. When we say that x is a limit of y's, we mean that the y's get closer and closer to x. Topologically, that doesn't happen at all with a sphere.

As far as local -- no, by local, I mean local stays just as small on a big sphere, but there's a lot more of it. That's why the local curvature can go to zero while the total curvature remains the same.

Yes, the local curvature approaches 0, and area which is approximately flat approaches infinity - while the ratio of that area to the total area does not change - but i'm not that interested in the ratio, it becomes undefined at infinity - as for what happens topologically... I suspect that is a matter of convention. Rather which conventional logic you choose to use... Or we're disagreeing on semantic grounds.

Your definition of a limit is correct, i'm thinking of the dynamic structure, i would love to be able to describe a continumn from negative curvature through 0 curvature to positive curvature.... (can you have an infinite spherical space - simply connected, isotropic, homogenous, continous, ?compact? OR a fininte flat/hyperbolic space?)

The limit as curvature approaches 0 (from positive) is radius of curvate approaching infinity, right?