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While reading Mathprof's entry "metacompact", I was surprised to notice that the word "point" had no link. I was curious enough to look at the TeX code and was surprised to find that he had not supressed a link.

After looking around on PM to be sure that "point" really was not an entry, I decided that it would be a good idea to have a definition of it. This is my humble and modest beginning. I am quite sure that the word "point" has many different definitions in many different areas of mathematics, and I would appreciate it if you would add them. Thanks.


A point is defined in my incidence geometry entry. But I felt that the usage of a point is so pervasive.. and not restricted to math only, that I did not define it as a "PM concept". However, I think it's a good idea that you are making "point" a stand-alone entry, allowing it to have multiple definitions, and making it world-editable, so everyone who wants to put his/her two cents in it can do so!


Well, just to start the discussion, why not define it as an element of a topological space? (Or, more formally, if we think about a topological space as a set with some extra structure, an element of that set).

One would obviously include many examples of the different usages, but off the top of my head, I can't think of any instances of using the word point that isn't a specific example of the above.


A point is not necessarily embedded in a topological space. One of the most interesting applications of the object "point" come from the so-called finite geometries, which have been studied to great extent. (One considers a geometry with some finite number of points and some set of lines between them; e.g. these can be constructed so they become counterexamples to well-known axioms).

So I think that "point" should be defined in more generality. However, in my opinion, this is a truly difficult definition... Maybe one should follow Euclid: a point is that which has no part, a line is breadthless length, ...


I don't really know what I'm talking about here, but...

What's "a geometry"? Aren't you just using to mean a finite topological space?


Not quite. A geometry is a set with some kind of an incidence relation (reflexive and symmetric) defined on it. Of course, there are more ingredients that can be added to this set so the concept of a point (and a line, etc...) becomes clear. So I think the two notions (geometry vs topology) are different.

Just to add to what CWoo has already said, in a "finite geometry" one defines a set of "points" (whatever those may be) and a set of lines (defined as a set of pairs of points between which we claim the existence of one or more lines). I guess one could add some sort of boring topology, but I don't think a topology has much meaning in a set with only finitely many points. Besides, it is not the topology that one wants to study but rather the possible geometric properties of these finite settings. Some of these finite models can serve as small scale models of more complicated geometric objects.


Here are two suggestions that I feel ought be considered when definitions are created by committees developing encyclopedias such as PlanetMath and Wikipedia. Basically I believe that any definition worth writing down should be constructed so as to be helpful to a reader's learning process:

1) All too often, the soldiers play follow the leader, giving the most credence and definition space to the perceived smartest contributor, else to the most advanced and abstract statements of mathematics that can possibly be made about a particular definition. Consider the POV of a young but enthusiastic student of mathematics who comes across the term point, a situation almost guaranteed to occur rather early in any math student's experience. Shouldn't that less formally trained, but eager to learn, person be entitled to first read a simple motivational definition, an informal, intuitive definition, such as that given by Euclid? Perhaps such an intuitive definition could be stated at the beginning of all of the basic math definitions, but especially the simplest math objects, and given with some amount of introductory discussion surrounding, say in this case, perhaps Euclid's dilemma in forming his definition of point, or some considerations that one needs to make when trying to specify/define what a mathematical point actually is, versus what a point is in the real world. This short but superfluous verbiage could actually help a new student gain an AHA take away, in learning that the definition of the supposed simple term point is actually not as simple as might be at first thought, and not as complex as suggested by the list given above in isolation. Without any motivating rationale suggesting why the definition of point should be considered at all interesting, this definition is lifeless and thus a detriment to learning.

2) Frequently, the term point itself is used in comments about the definition of the term point, leading to circularity in thought. This applies to most definitions, and should be avoided as much as possible.

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