[longshadow]

Eg. Does Topology describe in a Euclidean way, by use of discreet cumulative adjacent planes a non-euclidean form?

As best I can understand you question, no; I don't think that's it.

In topology, the word "manifold" refers to the subset of topological spaces that are locally Euclidean. IOW, it is a category of topological spaces that satisfy an additional constraint: they behave like Euclidean spaces if you don't stray too far -- IOW, Euclidean Geometry might work on short distance scales, but not between points in the space which are more widely separately.

The surface of the Earth is an example: locally, we can't tell that we aren't on a Euclidean plane, but if you zoom out far enough and watch ships sail over the horizon, you begin to realize it isn't really flat. Euclidean geometry doesn't work on large distance on the surface of the Earth. Shortest paths between two points are "Great Circle " routes instead of straight lines, but on small distance scales, the Great Circles become indistinguishable from straight lines connecting the two points. That's "locally Euclidean" but globally non-Euclidean.....

Does that help?

[Doctor Stochastic]

Just as an example; one can take the Cartesian product of two Brownian motions as a two-dimensional object that isn't a manifold (I think, maybe I'm wrong.) The x coordinate is continous but not differentiable for any y and vice versa.

[tcuoohjohn]

Yes..I think it does. Using your notion of Locally euclidean and the globe. Given that the horizon is at 12 miles or so and it appears planar. If you moved to that exact spot on the horizon you would move the next adjacent locally euclidean space.It's still locally euclidean ( planar) in all directions but globally non-euclidean. (curvilinear)

Close?