Topology/Topological Spaces

Topology
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In this section, we will define what a topology is and give some examples and basic constructions.

MotivationEdit

In Abstract Algebra, a field generalizes the concept of operations on the real number line. This general definition allows concepts about quite different mathematical objects to be grasped intuitively by comparison with the real numbers. Likewise, the concept of a topological space is concerned with generalizing the structure of sets in Euclidean spaces. Of course, for many topological spaces the similarities are remote, but aid in judgment and guide proofs. Interesting differences in the structure of sets in Euclidean space, which have analogies in topological spaces, are connectedness, compactness, dimensionality, and the presence of "holes".

If we begin with an arbitrary set, it may not be immediately obvious what is needed to imbue it with an interesting structure. One possibility might be to define a metric on the set, but as it turns out, requiring a metric is overly restrictive. In fact, there are many equivalent ways to define what we will call a topological space just by defining families of subsets of a given set. The properties of the topological space depend on the number of subsets and the ways in which these sets overlap. Topological spaces can be fine or coarse, connected or disconnected, have few or many dimensions.

The most popular way to define a topological space is in terms of open sets, analogous to those of Euclidean Space. (In Euclidean space, an open set is intuitively seen as a set that does not contain its "boundary").

Definition of a topological spaceEdit

Suppose we are given two sets,   and  , where   is a collection of subsets of  .

If   has the following properties:

  • the empty set and   are both in  ,
  • any (finite or infinite) union of sets in   is itself in  , and
  • any finite intersection of sets in   is itself in  ,

then   is called a topology on  . The ordered pair   is called a topological space.

This definition of a topological space allows us to redefine open sets as well. Previously, we defined a set to be open if it contained all of its interior points, and the interior of a set was defined by open balls, which required a metric. That is, we needed some notion of distance in order to define open sets. Topological spaces have no such requirement. In fact, the three properties given above— and them alone — are enough to define an open set. Our new definition is this:

  • the elements of   are called open sets.

Or, in other words, an open set is an element of a topology. So a topology is really a collection of open sets. The rules above are descriptions of how open sets behave: a collection of sets can be called open if the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. (The empty set is considered open by default). We also say that a set is closed if its complement is open. As mentioned before, a set can be both open and closed at the same time: the space itself is open, but since its complement (the empty set) is open, it is also closed.

As it happens, the properties we associate with open sets, which allow us to study many topological ideas (like continuity and convergence, which were defined earlier using open sets), are encoded entirely by the three properties described above, without any need for a distance metric at all. This is in fact a very abstract definition, using only the most basic ideas of set theory (subsets, unions and intersections), and it allows enormous flexibility in what can be studied as a topological space (as well as how something can be seen as a topological space; there are many different ways a topology can be chosen on a given set). This definition is so general, in fact, that topological spaces appear naturally in virtually every branch of mathematics, and topology is considered one of the great unifying topics of mathematics.

So, to recap: a topology on a set is a collection of subsets which contains the empty set and the set itself, and is closed under unions and finite intersections. The sets that are in the topology are open and their complements are closed. A topological space is a set together with a topology on it.

Some things to note:Edit

  • A set is only open under a particular topology. It does not strictly make sense to merely say that a set is open. In practice, however, the topology under consideration is usually clear from context. For instance, suppose we have the set  , and the set  .   is a topology on  , and one would say that the set   is open. This is inferred to mean that   is open under  . If we were given another topology  , then the set   would be open under   but not open under  . In this case it would be ambiguous to merely say that   is open. (Incidentally, it is not closed under   either, because its complement   is not open)
  • While the term topological space strictly refers to the ordered pair   (where   is a set and   is a topology on  ), often the topological space   is used interchangeably with the underlying set  , or the topology  . Which one is meant is usually clear from context. For instance, one might say "suppose   is a topological space and the interval   is a subset of  . Now, suppose   is open..." and so on. In this case (assuming the author has not made a mistake), it is meant that the topological space   is the pair  , where   is a subset of  , and   is a subset of  . In the case of the real numbers, usually the topology   is the usual topology on  , where the open sets are either open intervals, or the union of open intervals.
  • Infinite intersections of open sets do not need to be open. For example, consider open intervals of the form  . The intersection of these intervals for all positive integer values of   is the set  , which is not open in the real numbers.

Examples of topological spacesEdit

For any set  , there are two topologies we can always define on  :

  • The Discrete topology - the topology consisting of all subsets of a set  .
  • The Indiscrete topology (also known as the trivial topology) - the topology consisting of just   and the empty set,  .

Metric TopologyEdit

Given a metric space  , its metric topology is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as the set of all open subsets defined by the metric. We denote the topology   induced from the metric d with  

This forms a topological space from a metric space.

If for a topological space  , we can find a metric  , such that  , then the topological space is called metrizable.

The usual topology on the real numbersEdit

We can define a topology   on   by defining   to be in   if for every point  , there is an   such that  . We call this topology the standard topology, or usual topology on  .

The cofinite topology on any setEdit

Let   be a non-empty set. Define   to be the collection of all subsets   of   satisfying the following:

  1. Either  
  2. Or   is finite.

(In other words, the open sets are formed by removing a finite number of elements from  )

Then   is a topology on   called the cofinite topology (or "finite complement topology") on  . Further, this topology turns out to be discrete if and only if   is finite.

The cocountable topology on any setEdit

Let   be a non-empty set. Define   to be the collection of all subsets   of   satisfying the following:

  1. Either  
  2. Or   is countable.

(In other words, the open sets are formed by removing a countably infinite number of elements from  )

Then   is a topology on   called the cocountable topology (or "countable complement topology") on  . Further, this topology turns out to be discrete if and only if   is countable.

Sets in topological spacesEdit

Let   be a topological space. There are many types of sets we can define on  

  • The complement of a set A in X, denoted by  , is   (that is, the entire space except for A).
  • A subset   is called closed if the set   is open. Notice that the intersection of any non-zero number of closed sets is closed and the union of finitely many closed sets is closed.
  • Note also that a set can be both closed and open. The trivial examples are the empty set   and the entire set  , each of which is both closed and open. By definition,   is open, so its complement,  , is closed. But  , by definition, is an open set, so   is both open and closed.
  • A set   is called a neighborhood of a point   if it includes an open set containing  . In other words,   is a neighborhood of   if there is an open set   such that  


We now investigate some commonly occurring sets in the study of Topology.

DefinitionEdit

In a topological space, a   set is a countable intersection of open sets. A   set is a countable union of closed sets.

TheoremEdit

The complement of a   set is  , and vice versa.

Proof:
Let A be a   set and let  . Then A is a countable union of closed sets,   such that   is closed for all n. Then  . Since   is closed,   is open, so we have a countable intersection of open sets. Hence   is  .

The entirely similar proof of the other implication is left to the reader.

TheoremEdit

In any metric space, a closed set is a   set.
Proof:

Let X be a metric space and let  .
Define  . Observe that   is open for any n, and hence the union is open. Now our goal is to show that   to show that a closed set is the intersection of countably many open sets.

 :
Let  . Then   intersects A at some   which implies  . This is true for any n so  .

 :
Let   and  . Then   such that  . So   in A such that  , which implies  . Thus  .

Therefore   and is a   set.

TheoremEdit

In usual  ,   is a   set.

Proof:
Since   with the usual topology is a metric space, every singleton such that   is closed. Thus, we have a countable union of closed sets, and hence   is a   set.

ExercisesEdit

  1. Prove the following are topologies:
    • The discrete topology on any set.
    • The indiscrete topology on any set.
    • The cofinite topology on any set.
    • The cocountable topology on any set.
  2. Show that the cofinite (respectively, cocountable) topology on a set   equals the discrete topology if and only if   is finite (respectively, countable).
  3. Prove that a set is open if and only if for every element within the set, there is a neighborhood contained within the set.
  4. Show that the discrete topology is the topology induced by the discrete metric. (This is also a splendid way of remembering the discrete and the indiscrete topology)


Topology
 ← Metric Spaces Topological Spaces Bases → 

  (16:28, 31 March 2008 (UTC))