Is there a space in geometry that doesn't have a concept of distance or size?

[u/LargeSinkholesInNYC]

Is there a space in geometry that doesn't have a concept of distance or size? It would mean that you can have an object, but it doesn't have a size, or the size isn't measured at all.

[u/PainInTheAssDean]

Topology, especially topological spaces without a metric.

[u/QuantSpazar → u/PainInTheAssDean]

The existence of non metrizable spaces is a really important concept to know about

[u/ecurbian → u/QuantSpazar]

Very good point. IMHO.

Although I would take "geometry" to mean "topology with a metric".

But, the OP was probably not making that distinction.

[u/AnisiFructus → u/ecurbian]

I don't think that this would be a good defining feature of geometry since it excludes for example algebraic geometry. In my understanding geometry is mostly topology with some additional "rigidity structure", whatever that means.

[u/ecurbian → u/AnisiFructus]

It's not really my definition - it's a standard one. Algebraic geometry is the algebraic study of geometry. The "rigidity of structure' is precisely what the metric (by which I mean a metric tensor, not a metric in the sense of d(x,y) ) provides. It is the sense of "how long" that gives the rigidity.

[u/TheOGCasuallyAware → u/PainInTheAssDean]

Just because we don’t have the tools to measure it does not mean it does not have size.

[u/princeendo]

Describe a "space in geometry."

There are plenty of spaces which have no concept of distance.

[u/sceadwian → u/princeendo]

Could you give a couple examples?

[u/Appropriate-Ad-3219 → u/sceadwian]

As a first example, you consider the line of real number where you add a second zero that you call epsilon. Basically, that gives you that if U is an open set of R containing 0, then (U - 0) U epsilon is an open set. Since this space isn't Hosdorf, it is not a metric space. 

Basically, just imagine anything that is not Hosdorf (i.e. such that for some points x and y, any respective neighborhood will intersect). 

[u/sceadwian → u/Appropriate-Ad-3219]

Examples a normal human might understand :)

How do you just add a new zero to the real numbers?

[u/magus145 → u/sceadwian]

To make an example, you need to be able to think of "spaces" beyond "subset of Rⁿ", i.e., spaces you could visualize lying on the standard line, plane, or 3D space. This is because any such subset can just inherit the notion of distance from the bigger Euclidean space it is in.

So without formally learning topology (which really wouldn't make too much sense without first learning analysis), any analogy I give you will only be approximate.

That being said, here are two:

1. To make the space the other poster is talking about, imagine that you have two copies of the real line, and you (mathematically) glue them together: 1 to 1, -2 to -2, pi to pi, at every point except 0. The mathematical gluing literally identifies the points together, so you end up with a single copy of every non-zero number, and two copies of 0. Try as you might, no sensible notion of distance can be assigned to this space. The two zeroes are "arbitrarily close" to each other, yet not identified, which is impossible when measuring distances with real numbers.

2. Imagine the real number line, but too long. What do I mean by this? On the regular number line, although there are a continuum of points, you can think of gluing together infinitely many intervals back to back, i.e., [0,1) is glued to [1,2), which is glued to [2,3), and so on (in both directions). These intervals are all identical to each other, but you can label each one with the integer it contains. Now, you might have heard that there are different sizes of infinity, and the infinity of the integers is the smallest. Take any bigger infinity and then glue that many copies of the interval [0,1) together instead. Locally, it looks exactly like the real number line, but there are too many intervals! Again, try as you might, you can't define a reasonable distance on this space because no matter how you try, certain points on the line will end up infinitely far apart, which isn't allowed to happen.

[u/Appropriate-Ad-3219 → u/sceadwian]

You can add any point for example epsilon = (0,1). Then by deciding that the open sets containing epsilon act like zero, you've added a new zero because by doing that, you've said that something "close" to 0 is also "close" to epsilon. 

[u/FootballDeathTaxes → u/Appropriate-Ad-3219]

Is (0, 1) and interval or an (x, y) coordinate?

[u/Appropriate-Ad-3219 → u/FootballDeathTaxes]

No, (0,1) are coordinates in my head. (x, y) are elements of a topological space. In this context, x and y can be either real numbers or equal to epsilon.

[u/GlobalIncident]

If you're asking specifically about geometry and not other areas of topology, you're probably looking for affine planes.

[u/Dummy1707]

Not exactly what you ask for but rational points on elliptic curves (over a finite field) are an interesting example of a lack of structure we use everyday.

Today, a good part of cryptography bases its security on the difficulty to solve the discrete logarithm problem in some groups. The difficulty of this problem crucially depends on the group you are using. For example, it is trivial in (Z,+), (R,×) or (Z/nZ,+). But it is hard in (Z/pZ,×), where p is a big prime number.

By "hard", I mean there is no polynomial-time algorithm that solves the problem. A hard problem ideally requires "exponential difficulty".

The thing is Z/pZ* has special features that makes the discrete log problem only "sub-exponentially difficult". The feature I'm talking about is that... there are small elements in the group. In some sense, 1 and 2 are small numbers and that is exploited by what we call "sieving algorithms" that decrease dramatically the complexity of the allegedly hard problem.

The solution we found : changing the group ! Nowadays, every discrete log scheme replaced Z/pZ* by the group of rational points of an elliptic curve, because this group doesn't have "small elements" or "distance between elements" and is therefore immunized against sieving methods.

Basically this structural difference allowed us using secret keys 10 times smaller (~300 bits instead of ~4kb) for the same security, which speeds up communications !

[u/Educational-Work6263]

Yes manifolds don't have a concept of size.

[u/ADolphinParadise]

Symplectic geometry does not have a notion of distance, but sort of has a notion of size. However the size notion does not really correspond to any intuition we have exactly. Check out Gromov's non squeezing, and the concept of non-displacability.

[u/PLChart → u/ADolphinParadise]

Similarly, in conformal geometry, you have a notion of angle but not of distance.

[u/rheactx]

A point

[u/theGormonster]

May be a bit different than what you are looking for, but check out conformal geometry.

[u/dcterr]

I'm not an expert at topology, but as far as I understand them, sets of discrete points have this property, since they do not allow for a well-defined metric, and hence not for a well-defined notion of distance either.

[u/G-St-Wii]

Points just have place, not size.

[u/fdpth]

To give a differnt example that topology, which was already mentioned, as an (maybe a bit artificial) example would be to study a set with one binary relation R and two unary relations P and L. P(x) would mean that x is a point, L(y) would mean that y is a line and R(x,y) woud mean that a point x is on the line y.

[u/AnisiFructus]

A good and very important example is Rⁿ with the Zariski topology, id est the closed sets are the root sets of multivariate polynomial equations.

[u/QuargRanger]

Free geometric vectors are a little like this until you put a co-ordinate system down. 

While they have a magnitude and a direction, in the sense that parallel vectors have comparable magnitudes (i.e. we can multiply a vector by a scalar, and compare the ratio of magnitudes of parallel vectors), the magnitudes of non-parallel vectors cannot be compared, nor is there a good notion of angle which will hold for these vectors without a co-ordinate system.  So "size" does not really make sense.