2
$\begingroup$

I was reading Isham, Chris J. Modern differential geometry for physicists. Vol. 61. World Scientific, 1999. p.52

In the first chapter, he gives mathematical preliminaries that'll be useful for the rest of the chapters. There I came across this interesting text about open sets and their role in physics.

An important question in any topological space $X$ is the extent to which points can be distinguished from each other by listing the collection of open sets to which each belongs.

From the viewpoint of conventional physics, this is related to the idea that if $X$ represents physical space, then any real ‘object’ exists inside an open set. More precisely, it cannot exist as a subset of a closed subset unless this has a non-trivial interior. It thus seems plausible to argue that it is physically meaningless to distinguish between two points in $X$ if the collections of open sets to which they belong are identical. In the context of quantum field theory, this remark is related to the analysis by Bohr and Rosenfeld of the need to smear quantum fields with test functions that are non-vanishing on an open set.

One could say that all open sets are ‘fat’ whereas closed sets come in both thin and fat varieties. For example, a segment of a line in the plane is thin whereas a closed disc is fat.

Then the author defines what is a $T_0, T_1, T_2$ space, then he comments

Any topological space that represents spacetime must be at least $T_0$ at least if all its points are to have ‘physical meaning’ in the sense of being distinguishable by objects located in open sets.

I know one way to define topological space, continuous maps are in terms of open sets, but there also exist equivalent formulations in terms of closed sets for topology and continuous maps.

  • So why open sets are important in physics?
  • And what does it mean to put real objects in open sets?
  • How does one understand 'fat' and 'thin' sets?
$\endgroup$
0
6
$\begingroup$

The author is asking you to imagine a physical object - something extended in three dimensions, with a length, width and height. You can "put" this object into a topological space by taking $\mathbb{R}^3$ and consider "the object" to be the set of points there whose positions correspond to its actual position. If we have, say, a cube of 1 mm side length we would associate the set $\{ (x,y,z) \vert 0 \leq x \leq 1 \land 0\leq y \leq 1 \land 0 \leq z \leq 1\}$ with it, and this set is what the author calls "the object". That set is closed, but it is contained in many open sets.

The author's point is now that open sets "have volume" in that they can contain such "objects", while a closed set might not - a 2d surface is a closed set in 3d space, but it cannot contain a 3d object. This is what the author calls a "thin" closed set, while they call sets that have volume "fat". So if we are interested in representing phyiscal objects, it is more natural to think about open sets because they all can contain objects, while the closed sets can't.

A formal notion of volume for sets is that of a measure, and our intuitive notion of volume in $\mathbb{R}^n$ is modeled by the Lebesgue measure. Every open set has non-zero Lebesgue measure, but not every closed one has.

$\endgroup$
4
  • $\begingroup$ The only thing I can't understand is why the "object" which is a closed set can't exist as a subset of a closed set. Any arbitrary union and the finite intersection of a closed set is also closed. So I can write a family of closed sets that makes up the whole object. The closed sets can be put into closed sets of a topology. If I go along this line of thought I can see real objects also can stay inside closed sets. It's just the matter of defining the topology in terms of open or closed sets. So what is the "nontrivial interior" of closed sets am I missing? $\endgroup$
    – Galilean
    Aug 5 '20 at 5:40
  • $\begingroup$ @Galilean The point is not that it’s impossible to find a closed set which contains the object; it’s that a generic closed set may not have the capacity to contain an object at all (e.g. a closed disk in $\mathbb R^3$). In contrast, since all open sets must have nonzero volume, they provide a more natural notation of a container. $\endgroup$
    – J. Murray
    Aug 5 '20 at 7:39
  • $\begingroup$ @J.Murray how to define the volume of a set? $\endgroup$
    – Galilean
    Aug 5 '20 at 8:21
  • $\begingroup$ @Galilean I've amended the answer with a formal notion of volume. $\endgroup$
    – ACuriousMind
    Aug 5 '20 at 11:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.