A Digression on Topology (1)
Basics
With metric spaces we generalized the notion of distance; with topology we abstract things even further and characterize the notion of βarbitrarily closeβ in a qualitative sense without putting numbers on anything. It turns out that most or all the properties we care about for our pseudo-metric π on generalized premises are topological in nature, including dense sets, limits, and continuity; topology is the natural language for discussing these concepts stripped of unnecessary details.
Open sets in β
Topology revolves around the notion of open sets; we can define continuity, limits, and dense sets entirely in terms of open sets, without ever mentioning distances. The simplest and most familiar example of an open set, for the real numbers β, is an open interval (π, π): all numbers from π to π, excluding the end-points. In the standard topology on β, an open set is the union of any collection of open intervals; here are some examples.
The empty set β is an open set, as it is the union of an empty collection of open intervals.
(π, β) is an open set for any π β β, as it is the union of all intervals (π+π, π+π+2) where π is a nonnegative integer. (The intervals overlap to cover the endpoints.) Likewise, (-β, π) is an open set for any π β β.
β is an open set, as it is the union of (-β, 1) and (0, β).
ββ{π} (all real numbers except for π) is an open set, for any π β β. In fact, ββπ΄ is an open set for any finite set π΄.
If π and π are open sets in β, then so is π β© π. This is a consequence of the fact that the intersection of any two open intervals is another (possibly empty) open interval. E.g.,
\(\left(0,2\right)\cap\left(1,3\right)=\left(1,2\right);\)and when we intersect unions, we get a union of intersections, e.g.
\(\begin{align*} & \left(A_{1}\cup A_{2}\right)\cap\left(B_{1}\cup B_{2}\right)\\ = & \left(A_{1}\cap B_{1}\right)\cup\left(A_{1}\cap B_{2}\right)\cup\left(A_{2}\cap B_{1}\right)\cup\left(A_{2}\cap B_{2}\right). \end{align*}\)The union of any collection of open sets in β is itself an open set. This follows because the union of unions is itself a union. E.g., if π = π΄β βͺ π΄β and π = π΄β βͺ π΄β βͺ π΄β then
\(U\cup V=A_{1}\cup A_{2}\cup A_{3}\cup A_{4}\cup A_{5}.\)
The open sets in β give us a way of talking about points that are βarbitrarily closeβ to a point because for any π₯ β β we can find arbitrarily small open sets containing both π₯ and additional points nearby. In particular, the open interval (π₯-π, π₯+π), with π > 0 chosen arbitrarily small, contains both π₯ and additional nearby points.
Topological spaces
We can define dense sets on β, limits of sequences in β, and continuity of functions on β entirely in terms of open sets, and properties 1, 3, 5, and 6 above are all we need to prove many useful theorems about these concepts. This motivates the following:
Definition. A topology on a set π is a collection π of subsets of π, called the open sets, having the following properties:
β and π are open sets;
the union of any collection of open sets is itself an open set; and
π΄ β© π΅ is an open set whenever π΄ and π΅ are open sets.
The ordered pair (π, π) is called a topological space.
By repeated application of property 3 above we see that the intersection of any finite collection of open sets is also an open set. This contrasts with unions: the union of any collection of open sets is an open set.
Since weβre going to be talking a lot about unions and intersections of collections of sets, letβs clarify a bit of standard set-theoretic notation. If π is a collection of sets, then βπ is the union of all the sets in π. That is,
Likewise, βπ is the intersection of all the sets in π, i.e.,
We will often be interested in open sets containing some particular point:
Definition. An open neighborhood of a point π₯ in a topological space is any open set π containing π₯. Dropping the modifier βopen,β a neighborhood of a point π₯ is any superset of an open neighborhood of π₯.
In β, the open interval (0, 1) is one of the open neighborhoods of π₯ = 2/3, and the closed interval [0, 1] (which includes the endpoints) is one of the neighborhoods of π₯.
Examples of topological spaces
Example. The trivial topology on a set π is π = {β , π} . The only open sets are the empty set and π itself. If we think of any point π₯ β π as being described by the open sets to which it belongs, all points in π have identical descriptions, so this a maximally coarse-grained topology in which we donβt distinguish any point from any other. This notion suggests the following:
Definition. We say that two points π₯ and π¦ in a topological space (π,π) are topologically equivalent, or topologically indistinguishable, if π₯ and π¦ belong to the same open sets; that is, π₯ β π iff π¦ β π, for all π β π.
So for the trivial topology, all points in the space are topologically equivalent.
Example. The discrete topology on a set π is π = {π΄ : π΄ β π} , that is, all subsets of π are open. This is the opposite extreme from the trivial topology. No two distinct points π₯,π¦ β π, π₯ β π¦, are topologically equivalent. When you use the discrete topology on a set, you are saying that you have no notion of βpoints arbitrarily close to but distinct from π₯,β because the singleton set {π₯} is an open set that contains π₯ but no other βnearbyβ points. This notion suggests the following:
Definition. An isolated point in a topological space is any point π₯ for which the singleton set {π₯} is an open set.
So for the discrete topology, all points in the space are isolated points.
The intuition for this term βisolated pointβ comes from thinking about sets of real numbers. Consider the subspace topology for π = { -1} βͺ (0, β), the set of all positive real numbers plus one additional point, -1. The open sets in π are all sets of the form π β© π, where π is an open set in β. That extra point -1 doesnβt have arbitrarily close neighboring points; it is isolated from its nearest neighbors. Furthermore,
and so {-1} is an open set in the subspace topology for π.
For our next example weβll need the concept of open balls:
Definition. For any pseudo-metric space (π, π), π₯ β π, and π > 0, the open ball of radius π centered on π₯ is
When it is clear from context, we may omit the subscript π and just write π΅(π₯,r).
Assuming the Euclidean metric, which is the standard metric on ββΏ for any π, we have:
for β an open ball is just an open interval, which omits the endpoints;
for βΒ² an open ball is the interior of a circle, which omits the perimeter;
for βΒ³ an open ball is the interior of a sphere, which omits the surface.
Example. Any pseudo-metric π on a set π induces a standard topology on π that has as the open sets the union of any collection of open balls. (Weβll verify later that this satisfies the requirements for a topology.) The standard topology on ββΏ for any π is just the topology induced by the standard metric: the Euclidean metric.
The figure below illustrates four open sets in βΒ²; the sets themselves are colored light blue, and their boundaries are shown as dashed lines, indicating they are not part of the set.
The induced topology has these two useful properties:
Two points π₯ and π¦ are topologically equivalent if and only if π(π₯,π¦) = 0.
A set π is open iff for every π₯ β π there is some π > 0 such that π΅(π₯,π) β π. That is, each point in π is some positive distance within the interior of π; see the illustration below.
Closed sets
Definition. A closed set is the complement of an open set.
The complement is with respect to the entire space, so if we have a topological space (π, π) and open set π β π, its complement is
the set of all points in π that are not in π.
At first blush itβs not obvious why this is a sufficiently interesting concept to warrant a special name, but letβs look at some examples:
The set (-β, π) βͺ (π, β) is an open set in β, and its complement is the closed interval [π, π]. This is just the open interval (π, π) with the endpoints π and π included.
The set π = { (π₯,π¦) β βΒ² : π₯Β²+π¦Β² > πΒ² }, π > 0, is an open set1 in βΒ²; it is the exterior of a disk of radius π. Its complement is the closed disk
\(U^{c}=\left\{ (x,y)\in\mathbb{R}^{2}\colon x^{2}+y^{2}\leq r^{2}\right\} .\)This is just the open disk π΅((0, 0), π) with its boundary included.
In these examples the open sets consist entirely of their interiors, with the boundaries excluded, whereas the closed sets include their boundaries. We will see that these properties hold in general.
Definition. Let (π, π) be a topological space with π₯ β π΄ β π. Then π₯ is a limit point of π΄ if for every open neighborhood π of π₯, π΄ β© (πβ{π₯}) β β . That is, every open neighborhood of π₯ contains a point in π΄ other than π₯.
In the topology induced by a pseudo-metric, βevery open neighborhood of π₯ contains a point in π΄ other than π₯β is another way of saying βthere are points in π΄ arbitrarily close to, but different from, π₯.β This is a good way of thinking about what a limit point is in general. Note that, in line with this notion, an isolated point can never be a limit point of any set, since {π₯} is an open neighborhood of π₯, and {π₯}β{π₯} = β .
The concept of limit points gives us another way of characterizing closed sets:
Theorem. Let (π, π) be a topological space with π΄ β π. Then π΄ is closed iff every limit point of π΄ belongs to π΄.
This fact provides a way to minimally extend a set to a closed set.
Definition. The closure of a set π΄, written
or cl π΄, is the union of π΄ and the set of all its limit points.
Example. In a pseudo-metric space, the closure of the open ball
is the closed ball
Theorem. For any subset π΄ of a topological space, point π₯, and open set π:
cl π΄ is closed.
π΄ is closed iff cl π΄ = π΄.
cl π΄ is the smallest closed set that contains π΄.
That is, cl π΄ β π΅ for any closed set π΅ such that π΄ β π΅.
π₯ β cl π΄ iff π β© π΄ β β for every open neighborhood π of π₯.
That is, π₯ β cl π΄ iff every open neighborhood of π₯ contains points from π΄.
If π΄ β© π = β then (cl π΄) β© π = β .
As strange as it may seem, a set may be both open and closed.
Definition. A set in a topological space is said to be clopen if it is both open and closed.
For any topological space (π, π), both β and π are clopen. For the standard topology on β these are the only clopen sets, but other topologies may have many clopen sets. For example, in the discrete topology on a set π every set π΄ β π is clopen, since both π΄ and πβπ΄ are open.
It is open because π΅((π₯, π¦), π -r) β π for every (π₯, π¦) β π where π Β² = π₯Β²+π¦Β².