When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*.
But the topology we've talked about so far doesn't talk about shapes or surfaces. It talks about open sets and closed sets, about neighborhoods, even about filters; but we haven't touched on how this relates to our *intuitive* notion of shape.
Today, we'll make a start on the idea of surface and shape by defining what *interior* and *boundary* mean in a topological space.
We need to start with some concepts based on a notion of distance. In a topological space, the basic notion of distance is built on neighborhoods. For each concept, I'll start by describing it informally in terms of metric spaces, because that's where our intuitions of distance come from.
First, we need to be able to talk about what it means to be *arbitrarily close* to a set. In a metric space with metric function d, we can define the distance between a point and a set *S* by a function dS(x) = min({d(x,p) | p ∈ *S*}). So the distance from a point x to a set *S* is the distance between *x* and the *closest* member of *S*.
With that, we can talk about being *arbitrarily close* to a set. In a metric space, a point *p* is *arbitrarily close* to a set *S* if ∀ x ∈ ℜ > 0, dS(p) < x. That is, the distance from p to *S* is smaller than *any* finite distance. *(Note: originally, this said "x ∈ ℜ", ommitting the ">0". I was so busy focusing on the intuition that I neglected to make the formal part precise.)*
The set of all points that are arbitrarily close to a set *S* in a metric space are called the *closure* of S. That's usually written by *S* with a horizontal line over it by topologists; but since I can't find any good way to use that notation in HTML, I'll use the typical computer science notation for closure, and write *S\**. Intuitively, if *S* is an open set, then *S\** is the *closed* set containing *S* and its *boundary*.
What's the corresponding notion in an arbitrary (i.e., not necessarily metric) topological space? Suppose we've got a topological space (**T**, τ); and a subset *S* of **T**. The closure *S\** of *S* is the set of all points p where for every neighborhood of p, N(p), N(p) ∩ *S* ≠ ∅: that is, all points where one of their neighborhoods contains at least one point in *S*. It's the same basic idea as the metric space closure, but it's based on neighborhoods instance of distance metrics. In any metric space that's also a topological space, the topological and metric closures are the same. In fact, we can even formalize the idea that I mentioned up above about closed sets: a set *S* is a closed set if, and only if *S\* = S*. The closure of *S* is the *smallest* closed set containing *S*. *(Note: originally, this paragraph contained a typo: I used "∪" instead of "∩"; In HTML, that's "cup" instead of "cap".)*
It's a pretty easy to prove set of theorems that there are certain properties of a topological space involving closures. Given a topological space is (**T**, τ):
1. ∅\* = ∅. *(The closure of the empty set is the empty set.)*
2. **T**\* = **T**. *(The closure of the set all points in a topological space is the set of all points in the topological space).*
3. ∀ S ⊂ **T**: S ⊆ S\*. *(Every subset of **T** is a subset of its closure.)*
4. ∀ A,B ⊂ **T**: (A ∪ B)\* = A\*∪B\*. *(The closure of a union of two subsets of **T** is union of their closures; closure in invariant over set union.)*
5. ∀ S ⊂ **T**: S\* = S\*\*.
*(Taking the closure of a closure doesn't do anything.)*
Now that we can talk about the closure of sets in a topological space, we can move on to the definitions we really care about: interior and boundary.
In a *metric* space, we know what the interior is, intuitively. We can define any shape in a metric space by the combination of unions and intersections of *open balls*; so we can define interior for open-balls, and go from there. If B(p,x) is an open ball of size x around a point p, then a point z is in the *interior* of B(p,x) if d(z,x) < p. The extension of this over unions and intersections works exactly as your intuition would predict; the interior of a set in a metric space matches your intuition of *inside*.
What this means is that the interior of a set is a kind of *opposite* of the closure: the closure of a set S was the *smallest* closed set that includes S; the *interior* of a set S is the *largest* open set that is included by S.
And that's exactly the definition that we can if we expand to topological spaces: in a topological space (**T**, τ), a point p is in the interior of a set *S*⊂**T** if and only if *S* is a neighborhood of p. Work out the set of points p for which *S* is a neighborhood, and it's the *largest* open set included by *S*:
Given a subset *S* of a topological space (**T**,τ), the *interior* of *S* is the union of all open sets contained in *S*. The interior of the set S is often written Int(S).
*(Note: the original version of the above three paragraphs contained errors in the phrasing; they originally said "the interior of a set S is the largest open set that includes S, where it should have been the largest open set that *is included by* S.)*
And last but definitely not least; the *boundary* of a set *S* in a topological space (**T**,τ) is the intersection of the *closure* of *S* and the closure of the complement of S:
Boundary(S) = S\* ∩ (S-1)\*
Once again, if we were to look at this in a metric space, it's *exactly* what our intuition would call the boundary or surface of S. The surface of a sphere is the set of points which is arbitrarily close to both the *exterior* of the sphere and the interior of the sphere. It's the set of points that forms a *boundary* between the interior and the exterior.
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I think instead of "Work out the set of points p for which S is a neighborhood, and it's the largest open set including S" you meand "Work out the set of points p for which S is a neighborhood, and it's the largest open set included in S".
I think you mean it the other way around: in any metrizable topological space, the topological and metric closures are the same (and the metric closure doesn't depend on your choice of metrization).
Good math, Bad math. This is the bad math part, right.
Your distance between a point and a set should use infimum rather
than minimum. Using minimum the distance between 0 and the positive real
numbers is not defined (there is no "closest member".
Perhaps this was an oversight.
Next you said a point p is arbitrarily close to a set
S if for all real x, dS(p) <
x. Not sure how you've defined distances, but here you must be
allowing negative distances, or perhaps no points are arbitrarily close
to any set.
Correct definition: A point p is arbitrarily close to
a set S if for any positive real number x there is a
point s in S such that d(p,s) < x.
You never defined boundary other than to say that
S* is the (sic.) closed set containing S and
its boundary. (Isn't the entire topological or metric space a closed
set that contains S?). Perhaps you want to define the boundary here
to be the set of all points arbitrarily close to both S and its
complement. You get to this in a way in the end, with the intersection of closures. But of course there were problems with arbitrarily
close to begin with. Later you do say (correctly) that the
closure of a set is the smallest closed set contining S. In
fact since the intersection of arbitrarily many closed sets remains
closed, you can just define the closure to be the intersection of all
closed sets containing S. At any rate you can't use closure to define boundary and boundary to define closure.
You mixed up intersection and union in the definition of closure for
a general topological space. Perhaps this was a typo.
Your definition of interior is screwed up. The largest open set
containing any subset S of a topological space T is
T itself. Surely you mean the largest open set contained in
S. Again, since the arbitrary union of open sets is open, you
could just define the interior of S to be the union of all open
sets contained in S (you mention this later). Alternatively,
since you seem to be sticking to a definition regarding the individual
points, you want the interior to be the set of all points in S
having an open neighborhood entirely contained in S.
Randy:
Comments like yours are the kind of thing that make me wonder just why I spend so much time writing this blog. I don't mind corrections; in general, I welcome them. I *do* mind the obnoxious snide attitude.
<Mark C. Chu-Carroll>
From "Obnoxious Answers to Obnoxious Questions"
Posted on: September 6, 2006 2:29 PM, by Mark C. Chu-Carroll
Question: Why are you so rude? There's no need to call people stupid, or idiotic, or ridiculous just because they made a mistake.
Obnoxious Answer: I'm rude because it's fun.
Question: How would you feel if someone did a slam job on you like you've done to so many people?
Obnoxious answer: I'd get down on my knees and cry about how unfair the world is.
</Mark C. Chu-Carroll>
I took this as sarcasm, mixed with a bit of obnoxious snide attitude.
<Mark C. Chu-Carroll>
Serious answer: It's happened. People have picked posts on this blog, and used them to attack me. I've been called everything from a good for nothing claptrap nobody to a child molester. Sure it pisses me off. But that's life. In the end, truth wins out. If what I'm saying is right, then the fact that a couple of yahoos called me some rotten names is no big deal. If what I'm saying is wrong, then I deserve the abuse.
</Mark C. Chu-Carroll>
I have no idea what you do with children and am making no accusations. Your last sentence sort of says it all though, doesn't it.
Randy:
Thank you. I suspected that's where you were coming from. You're welcome to post whatever random shit you want in the comments, but don't expect any more acknowledgement.
Have you ever noticed that recent studies indicate that nucleic acids - through introns and exons ['plasticity' surgery?] - do not follow the [human] rules of topology?
Yet the various shapes and surfaces produced by nucleic acids can be represented as geometric or mathematical objects.
Rather nucleic acids appear to use attractor v dissipator chaotic gaming bifurcations.
Perhaps differential geometry is like a phenotype of energy interactions which in turn are like a genotype?
This may mean that degrees of freedom [dimensions] are more energy than spatial related?
There's something about this which is confusing me. I guess this is the best way I can think of to phrase it:
Let's say I've got a topology T, and a set U which contains exactly one single point from T.
Is U an open or a closed set, or neither?
What is the closure of U? If U is open, what is its boundary, if it has one? If U is closed, what is its interior, if it has one?
I'm having trouble understanding the idea of the boundary around a point (or even a finite set of points) in a continuous space, and this "the distance is smaller than any finite distance" terminology baffles me because it seems to admit the existence of infinitesimals. What am I missing here?
Coin,
Keep in mind that a topology T is a collection of subsets of another set, say X. T thus doesn't really contain "points" of X, but rather, elements of T are just subsets of X. I think you're trying to ask: If I have a topological space (X,T), and U={x} is a one point subset of X, is U open/closed/etc?
The answer is that it depends on the topology T you've given to X. In the discrete topology (T consists of all subsets of X), U would be both open and closed. In the indiscrete topology (T consists only of X and the empty set), U would be neither open nor closed. There are tons of topologies on X, and these properties of U will change depending on the one you pick.
So, it's hard to say much about the interior or closure of U without giving more information about the topology. In the discrete topology, U*=U, but in the indiscrete, U*=X, to use the notation for closure from above.
You are right that when we say a distance is smaller than any positive real number, we're looking at something infinitesimal in size. However, in order to even talk about things like size or distance, we need to have a metric, something an arbitrary topological space need not have. Our intuition about distance doesn't apply.
I'm not sure what you mean by continuous space, but perhaps what you have in mind is the real line, R. R has a handy metric: for x,y in R, d(x,y)=|x-y|. This metric induces a topology on R that's familiar: a set U in R is open iff for every x in U, there is an open interval (a,b) for which x is in (a,b) and (a,b) is contained in U. Using this, one can show that a single-point set {x} in R is closed but not open, that its interior is empty, and that its boundary is {x}. As before, though, this is determined by the topology induced by the Euclidean metric; it may change in other topologies.
One question, and one request:
In point (3) you say: "â S â T: S â S*. (Every subset of T is a subset of its closure.)" I'm wondering why these use â and â respectively. How could it be that S â T and yet S = S*?
And the request is this: When I learn a topic like this I like to have a "classical" example in mind... for topology it's basic Euclidean geometry. I use that for my first visualization (eg: I can visualize an open set as a region that's fuzzy because it doesn't have its surface). And then I want to read the formal definitions and theorums. But I find that my intuition doesn't really kick in until I also consider one or two pathological cases. After all, the general definitions are written to extend the basic concept into more general cases, and I can't understand the motivation until I have an idea of where we're headed.
So my request is, could you provide some examples of "pathological" (or at least non-trivial) cases where we would apply these topological notions?
And PS: Thanks for the WONDERFUL blog!
Michael,
As to your question about closures, any closed set is its own closure. Since S* is the smallest closed set containing S, if S itself is already closed, it follows that S*=S.
Regarding examples, perhaps the easiest thing to do is to look at a 3-point space, say X={a,b,c}. X has a couple of different topologies; 6, I think, up to symmetry. For instance, T={{},{a,b,c},{a,b},{b,c},{b}} should satisfy the axioms of a topology. (It might be helpful for you to look at other subsets of subsets of T and decide whether or not they are topologies.)
Now that we have a topological space (X,T), we can then discover things such as: {c} is closed in (X,T), {b} is open, {a,c} is closed. The interior of {b,c} is {b,c}, {a,b}*={a,b,c}, and the boundary of {a,b} is empty.
You can find all these things for every subset of X, given that you equip X with a topology. However, the space is so simple and so far removed from standard Euclidean space that it may help you get a handle on the level of abstraction topology requires.
Coin:
bbs mostly beat me to the answer.
The main trick in a lot of topology - particularly the kinds of algebraic topology that interest me - is that topological spaces are abstract, not concrete geometric spaces. The structure of the space is defined by the topology - which itself defines the open sets in a particular topological space. A *point* in a topological space isn't necessary a point as you'd think of it in, say, ℜ2, the real cartesion plane. A point in a topological space is just *any* object that's a member of a family of sets that has the correct properties to define a topology.
So a set of a single point could be open, or it could be closed, depending on the definition of the space. If it's closed, then it's closure is itself. If it's open, then the value of its closure is dependent on the properties of the topology.
Finally, metric spaces do have something like infinitessimals in terms of "distance". The idea that the "smaller than any finite distance" is trying to capture is the idea of *the single layer of points adjacent to* the outermost points in an open space. Think of the filled shape in ℜ2 defined by x2 + y2 < 1. It's a solid circle. But it doesn't include the outer line around the circle, the line x2 + y2 = 1. So how can we talk about that outer line surrounding the filled circle? It's the points at *exactly* x2 + y2=1. It's the set of points *directly adjacent* to members of the filled circle, but not inside of it. If you take any line crossing through (0,0), and follow it outward in either direction away from the origin, the outer circle contains the *first* point that the line encounters *after* leaving the filled area. But without using some construct that uses the concept of an infinitessimal, you can't express that. After all, *by definition*, in ℜ2, given any two points p and q, there's an infinite number of points between them. But if p is on the boundary of the circle, and q is in the interior, then anything *between* p and q is in the interior.
That's what the infinitessimal construct is expressing.
Michael:
The ⊂ in S ⊂ T could have been a ⊆. But even with it written as is, there are plenty of cases of subsets of T that are there own closure: by definition, *every* closed set is equal to its own closure, and most closed sets are not the set of all points in the topological space.
I'll try to come up with some pathological examples if I can. The way that I try to think when I'm writing is to ground things as well as possible in intuition. For the way that I think, that tends to lead me away from the pathological cases, because they're harder to explain intuitively. But I'll try to see if I can find cases that push the limits without blowing the intuitive sense I'm striving to create.