Not enough people on the internet appreciate math

Even on Path, and this is one of the most math-heavy games in the mainstream style.

Concepts in this post should all be covered in a basic topology class. Some you might even find in set theory, wouldn't know, never took that.

The question was raised, what is the definition of a "build"? The relevant part of my answer is below.

--
Observations:
- Path build n-space is discrete. It's also closed and contains a (very large, but) finite number of elements.
- Passive tree major or minor changes have some effect sometimes. Big nodes are game-changing, small nodes have cumulative effects, but the only logical way to resolve this is for no nodes to be valueless.
- Skills are different. Support combinations are different, to some extent. As above, there can be only one logical resolution to this. Both these spaces don't necessarily behave all that well if you pick the wrong definitions, so don't pick the wrong definitions.
- For the most part, rare item stats have less impact, and similarly with statball uniques like Kaom's Primacy. On the other hand, there are build definers like Facebreaker. Such items, then, each correspond to a single dimension in Path build n-space, or we can say that Path build n-space is the product space of several spaces, some of which are Boolean in nature and correspond to items. But even statballs have their impacts: physical, spell, both? How much total MF? And so observe: it makes sense to have each individual stat mapping to some interval in R, and the sum total of rares/statballs occupying some point in R^m.

That's...clearly not all different builds the way the term "build" is generally used, though a strict definition should define them that way, so let me introduce the term "setup". Each element of Path build n-space is a different setup.

Luckily for us, setups tend to cluster in different regions of each of the parent spaces that contribute to the product space. Just as an example, on the skills space, taking a subset of all setups which contain the active skill element LA, a lot of xMP, chain, WED, etc are found, while setups containing MPD and splash are conspicuously absent. (If you're wondering, I defined a compact R^6 where "support gem does not work with skill" means that element just maps back to 0 on that dimension. There's a lot of (LA,0,0,0,0,0) in it. Not necessarily the most efficient method, but I'll take that trade-off for the well-behaved-ness of R^6 any day!)

>> I believe it is possible to define a metric on Path build n-space. Just saying. This is important.

But going back to the idea of clusters, that's basically a finer - or coarser, sorry I'm on a timer here not going to figure that one out - version of each of the mentioned spaces. Each element in Path build cluster n-space is a different build. Alternatively, for each cluster choose a point near its centre* to be the element in the corresponding space, and thus define builds.

*this is why the metric was important

Problem solved, right? The only fuzzy bit is the clusters one, and I'm not happy with that either, but how else are you going to draw an arbitrary line? The second method might be better, as in "I'm playing a setup that's close to theWombo's build on Path build n-space".