Definition of "Up to" (homeomorphism,isotopy, etc), and Examples?

Question

I've tried googling this usage and understanding the results but I'm struggling to make intuitive sense of it. So my question is, what is the phrase "up to" understood to mean, and what are some prominent examples of this?

Answer 1

"X up to Y" means that X is an element of some set (or possibly proper class) but, instead of talking about elements of the set, we want to talk about equivalence classes of elements under some equivalence relation given by Y. Equivalence relations are ubiquitous in mathematics, hence so is the phrase "up to." For example, "classify compact surfaces up to homeomorphism" means we want to classify equivalence classes of $2$-dimensional compact surfaces under the relation "$S_1$ is homeomorphic to $S_2$."

Answer 2

You may be familiar with the integers modulo 3. This is a field with three elements. In fact, it is the only field with three elements, up to isomorphism.

But what are the integers modulo 3?

One might say it is the set whose three elements are

$$ 3\mathbb{Z} \qquad \qquad 1 + 3\mathbb{Z} \qquad \qquad 2 + 3\mathbb{Z}$$

(i.e the three equivalence classes of $\mathbb{Z}$ modulo the relation $x \equiv y \bmod 3$), with addition and multiplication defined the usual way.

However, one might say it is the set whose three elements are

$$ 0 \qquad \qquad 1 \qquad \qquad 2 $$

and addition and multiplication are defined by

$$ a \oplus b = \begin{cases} a + b & a + b < 3 \\ a + b - 3 & a + b \geq 3 \end{cases} $$

$$ a \odot b = \begin{cases} ab & ab < 3 \\ ab - 3 & ab \geq 3 \end{cases} $$

But wait there's more! One might instead say it is the set whose three elements are $\{ 0, 1, -1\}$, or occasionally one might even use $\{ 1, 2, 3 \}$.

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The point is, while technically all of these give different fields, the difference between them is superficial. Except in odd circumstances, there is absolutely nothing to be gained by keeping them all different in your mind; instead, they should all be regarded as the same thing.

This example is extra nice, because not only are they all isomorphic fields, but they are uniquely isomorphic -- there is only one way to choose an isomorphism between any two of them.