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I'm trying to teach myself some mathematics, and I want to see if I understand this concept correctly from what I've been reading.

(And just to be clear, this isn't part of any coursework, so I assume it doesn't go under that section for that reason?)

So, essentially, although there's a lot more to it, just in a quick summary, would it transfer the general idea correctly to say that a Topological Space is a set of points X, where each of the points has a function f(x)? Like, for example, a newtonian gravitational function of location,

[itex]f(x) = \frac{dv}{dt} = \frac{GM}{x^2}

[/itex]

If this were coupled with an open set of locations, would it be a Topological Space?

What if it's more generalized to 3-dimensional space, as,

[itex]F(x) = \frac{dv}{dt} = \frac{GM}{(x^2 + y^2 + z^2)}[/itex]

When would that be a Topological Space?

And also, one question about what Topological Space

*isn't*: If there were a portion of space where each point were assigned a vector for newtonian gravity, would that be a Vector Bundle instead?