Dimension

Sets

You will need to know about sets. Sets are easy (to get an initial grip on). Firstly sets are containers for things, sets of things. You can't have duplicates in a set, a thing is either in a set, or out of it, iut can't be in it twice, it's a with us or against us thing. Sets are often written as {a,b,c,d} where a,b,c and d are members of the set. {} is a set with nothing in it, it is the empty set. It is often written 0. Sets can contain sets. {{a,b},{a}} is the set containing two sets, firstly the set containing a and b and secondly the set containing just a. {{}} (or, equivalenty, {0}) is the set containing the empty set. That is different to the empty set, our set {0} contains something, one thing, the empty set, 0 contains nothing.

Sets can be joined together to make other sets. You can pool two (or more) sets to make a bigger set containing everything in any of them (duplicates are included just once, as ever), this is called the union and is written u. It works like add or times in arithmetic, you put two sets either side of it and get a third set out. So {1,2,3} u {1,4,5} = {1,2,3,4,5}. The other important operation is intersection. Intersections contain only the things in all of two (or more) sets. If it's not in one of the sets you start off with, it's not in the result, it's written n. So {1,2,3} n {1,4,5} = {1}.

A subset just means a set containing just some (or none) of the members of another (usually bigger) set. The empty set is a subset of every set. Every set is a subset of itself, the subset containing all the members! These two slightly cheaty subsets (all and nothing) are called the trivial subsets, the others are called the non-trivial subsets.

Topology

Here's a formal definition of a Topology. A topology is a set and some subsets of it, in fact a set of subsets of it. The set is called X and the set of subsets T. For this to be a topology there must be some properties shared by T and X.

• First, X and 0 must be in T. So both the trivial subsets are in T. So if X is {a,b,c}, then T must contain {a,b,c} and {}.
• If a group sets are in T then their union must be in T too. So if T has {a} and {b} in it, it must have {a,b} in it too.
• If two sets are in T then their intersection must be in T too. So if T has {a,b} and {a,c} in it, it must have {a} in it too.

If all these things are satisfied, then we say that X and T make a topology. T={{},{1,2,3},{2,3},{2}} X={1,2,3} is a topology, check it out to make sure you understand what's going on.

What does this have to do with space and shapes? What a Topology, defined above, does is help provide a very fluid idea of space without assuming too much or involving handwaving, which always causes trouble. The problem is, it seems so divorced from reality, it's difficult to see where we're going. Here's a brief explanation just to reassure you. X in the above is a space. Imagine, in a normal space, each of its members being a place in that space, a point, a dot. If a space is inifitely divisible, X will have infinitely many members, one for each place. If X isn't infinitely divisible, like a chess board it will have a finite number (64 for a chess board).

Open Sets (aka Open Intervals)

If X is a place, then T is all the open sets in that space, that is all that an open set is.

To help you get a grasp of what this means, think of all the possible intervals on a real line (or a finite number of squares, if you'd rather), which exclude their endpoints. Now consider the combinations of these intervals, so things like Between 1 and 3 (exclusive) or 7 and 12 (exclusive): that (intuitively) is an open set, a group of open segments. Note that it can, in general, have gaps in, between the intervals, that's fine. It's clear why T is the set of open sets by the way it was defined. If two things are open sets then their combination clearly is (the union rule), if two things are open sets which overlap then the intersection (being the things in both) are also open sets.

For example, in the union rule between 1 and 3 exclusive and between 7 and 12 exclusive are both open sets, so clearly the two things together are: between 1 and 3 exclusive or between 7 and 12 exclusive. For and example of the intersection rule, consider that last open set intersected with between 2 and 10 exclusive. The intersection is just between 2 and 3 exclusive and between 7 and 10 exclusive, which is another open set.

The opposite of an open set is a closed set which includes its endpoints. They're perfectly fine and respectible things, but not as important as open sets.

Open Cover

An open cover is a collection of open sets (that is, members of T from a topology X,T). These sets when unified together, a particular set being covered is a subset of a union of those sets. That's confusing. What it means is, given a whole group of open sets, all unified together to create an uber-open set, then the set being covered is contained within that uber-set. So the open sets between 1 and 4 and between 3 and 9, they together cover the open set from 2 to 6 (all exclusive of endpoints of course!). There's more than one way to cover each open set, of course. And as the union of a family of open sets is an open set itself, then every open set does, in-fact cover itself!

Refinement

Consider a cover of some open set, some higgledypiggldey set of overlapping open sets that between them cover some open set. A refinement is a different cover of the same set. This refinement, though, is more refined because each of the open sets used to cover the set is a smaller version of the original cover (or perhaps the same size). So whilst 1 to 4 and 3 to 9 covers 2 to 6, a refinement of that cover is 1 to 4 and 3 to 8, each element in the second cover being one in the first cover, but perhaps a bit smaller, perhaps the same size, but certainly not larger in any way. Formally a refinement is just a cover each of whose elements is a subset of another cover.

A refinement of a cover of a set depend not only on that covering, but upon the set which it covers. One refinement which makes a covering smaller will be acceptable for some sets we are trying to cover, but not for others (beacuse they cause a bit of the set to be uncovered).

Lebesgue Covering Dimension

At last! The Lebesgue Covering Dimension concerns finding a particular kind refinement of every possible cover you might imagine. That refinement needs to have this special property: for every point in the set to be covered, it's in no more than a certain number of the sets used to cover it in the refinement. We try to get that number as small as possible. That number is one more than the Lebesgue Dimension of the space.

That means that, for example, a line (which we are pretty sure is one-dimensional) needs, in general to have points in upto two covering sets. Why's that? Why can't we get away with them being in just one? Well, consider a big segment in the set to be covered, one that we cover with two intervals that overlap a bit. What we're trying to do is to shrink each of the covering sets so that points aren't unduely shared between them. We can do that a lot, but there always has to be a tiny bit of overlap, at least one point shared between those two lines. Why? Because the open sets are exclusive. We can't just jam them end-to-end because their endpoints aren't included, we have to include an overlap that we can make very tiny, but can never get rid of. Those few points will be in two cover-sets, so the Lebesgue dimension is one.

Consider now an open set that's a square. Again, think of a really big suqare that's covered with fragmentary squares making a kind of collage that, together, blankets the square completely. Now, we can shrink the squares of the collage, but we need to leave overlap at the edges, like with the line, so tiny bits of the square have to be in two. Those points along the edges have to be in at least two covering sets. But what about the region at each of the corners of a covering collage square? They're in the set that they're near the corner of, but they must also be in the collage square that overlaps the vertical edge of the square (because they're near the vertical edge) and in the collage square that overlaps the horizontal edge, they need to be in three collage squares, the square is two (one less than three) dimensional.

You can think of this like coloured gels (transparancies) on a light-box. You cover the light box and then slowly magically shrink them) until there's almost no overlap. But in the very corners of each square (the corners that are on the light-box, at least), there will be little patches of light that have been through three gels. Sure it's possible to think of a covering that doesn't have any such points, like a bloody gigantic gel that covers the whole light box, but that's not the deal. Given an existing cover that an evil genius has set up to thwart you and given only the special power of shrinking a square, the best you can hope for is little bits covered by three gels.

You might argue that four squares can meet at a corner, meaning that some points have to be in four covering sets. The thing is, you can always shrink the overlap a bit more. And if you're careful about how you shrink it you can turn this four way super-corner into two three-way normal corners, for example by shrinking diagonal squares.

Needs a lot of work -- Dan

This is intended to be a semi-formal introduction to the idea of classical Dimension for the intellegent non-specialist in set-theoretic terms. I am not an expert on the subject, please do correct. We need to do something about the lack of maths formatting in this Wiki -- Dan

There's a latex2html patch at [1]. I expect it's probably a security hole. Does LaTeX have a 'safe mode'? MHF

The business with the refinement of an arbitrary open cover is there as a way of saying 'small open sets'.

That is, you'd like to say that the dimension is determined by the number of overlaps required by a cover using arbitrarily small open sets, but topological spaces don't have a natural definition of small. So you invent an opponent who gives you whatever kind of sets sie feels like, which is enough to force you to cope with the small ones.

If you do have a notion of size of an open set (as you do in a metric space), you can use that and get the same answer for the dimension. -- MHF

If I understand you right (I only learnt this today!) you're saying that what we'd really like to say is given arbirtarily tiny sets how can we build those up to make lines, squares, cubes, and so on, how many overlaps do we need? As tiny ones are necessarily easier than big ones (big in the sense of a superset of tiny ones) because they have fewer of these awkward edges, we just let the evil adversary either choose (what we intutively understand as) tiny ones, or else make our life easy, and then extrapolate that for arbitrarily evil evil? Is that right? If so, I'll integrate that into the main text. --Dan (Thanks)

Tiny sets are harder than big ones -- that is, you might need more overlaps if you use tiny sets. This is because if you use small enough ones, they will end up being smaller than the size of the most complicated bit of the space.

If you have a line with a two-dimensional blob on it, it will look one-dimensional if you're allowed to use tiles that are bigger than the blob. But with small enough tiles, you'll have to notice it's really two-dimensional. Come to that, for any space, if you're allowed a big enough tile you can cover the whole thing in one go and make it look like a 0-dimensional point.

Cool. I get you now. I want to move this eventually to LebesgueCoveringDimension and create CapacityDimension? and InformationDimension? to cover things fractal. I can link in with those subjects by mentioning such concerns.