In this post we discuss the Euclidean topology of the real line as a way of motivating the notion of topological space. We also contrast the Euclidean topology with the Sorgenfrey Line, another topology that can be defined on the real number line. In the comparison of the two topologies, we focus our attention on continuous functions. For some (the author of this post included), exposure to the Euclidean topology (especially the real line) is a gateway to the study of general topology.
The Real Line
Let be the set of all real numbers. For any two real numbers and with , by the open interval , we mean the set of all real numbers such that . Let be a subset of . The set is said to be an open set if for each , there is an open interval containing such that . A set is said to be a closed set if the complement is an open set.
By definition, all open intervals are open sets. The set is an open set. Note that it is the union of two open intervals. The set of all positive numbers is open. Note that .
Let be the set of all subsets of that are open sets. The set is called a topology. To distinguish this topology from other topologies that may be defined on the real line, we call the topology just defined the Euclidean topology or the usual topology on the real line. We have some observations about open sets in the real line (stated in Theorem ).
- Both and the empty set are open sets.
- The union of open sets is an open set.
- The intersection of finitely many open sets is an open set.
Often times we work with subsets of the real lines. A few examples are: , , , . These are the set of all nonnegative integers, the set of all rational numbers, the set of all irrational numbers and the unit interval, respectively. Given a subset , can also have a topology. The open sets in the subspace are inherited from the overall real line (i.e. they are simply the open sets in the overall real line but points not in are excluded). For example, for each nonnegative integer , the singleton set is obviously not an open set in the real line. However , where , is an open set in the space . This is because we have:
Since every single point in is an open set, the space is said to be a discrete space. An open interval in is simply (i.e. the set of all rational bumbers with ). Note that every open interval contains infinitely many rational numbers. Thus is not a discrete space. For the unit interval , an open interval containing the point is where . Likewise, an open interval containing the left endpoint is . In general, for the subset , the open sets are generated by the open intervals of the form . The set of all open sets just defined for the subset is called the relative topology for since it is inherited from the topology for the overall real line.
The notion of continuous functions is a topological one. For some students, the notion of continuous functions is introduced in courses such as calculus and elementary analysis where the -defintion is used. We consider continuous functions from a topological point of view.
Let’s see how continuous function is defined in a typical calculus text such as . The function is continuous at if . For the definition of , we have the following from :
The number is the limit of as approaches provided that, given any number , there exists a number such that for all such that .
Let and let be a function. The function is said to be continuous at if for each open interval containing , there is some open interval containing such that . The function is said to be a continuous function if it is continuous at every .
Of course, definition is equivalent to defintion . In definition , if is the midpoint of the open interval , then corresponds to the in definition . Likewise is the in definition .
Defintion is wedded to the Euclidean metric in the real line. When considering functions defined on a higher dimensional Euclidean space or another type of spaces, defintion will have to be rewired so to speak. It also feels cluttered, not to mention being confusing to the typical students in a calculus class. Defintion , as we will see below, is applicable in a wide variety of settings. For example, when the topology changes, just replace with the new notion of open intervals in definition . We have the following characterizations of continuous functions:
Let and let be a function. Then the following conditions are equivalent:
- The function is continuous.
- For each open set , the inverse image is an open set in the domain space .
- For each closed set , the inverse image is a closed set in the domain space .
Some familiar examples of continuous functions include polynomial functions with real coefficients, trigonometric functions such as and as well as exponential functions such as and logarithmic functions such as .
The following function is not continuous at .
The following function is not continuous at every .
Let . Define by . The function is continuous at every . Thus it is a continuous function (as a function defined over ). However, it cannot be extended to a continuous function over the entire real line.
The Sorgenfrey Line
We now modify the definition of open intervals to be of the form where . In other words, the open intervals include the left endpoint. As before, a set is open if for each point , there is an open interval containing such that . Theorem still hold true with this definition of open sets defined in the real line. Let be the set of all open sets generated by the open intervals . The set is called the Sorgenfrey topology of the real line. Thus we have two topologies defined on , and . To avoid confusion, when we discuss the both types of open sets at the same time, we refer to the open sets in (defined by the open intervals ) as the Euclidean open sets or the usual open sets. We refer to the open sets in (defined by the open intervals ) as the Sorgenfrey open sets.
How are the Sorgenfrey open sets different from the usual open sets? First off, every usual open set is a Sorgenfrey open set (i.e. ). To see this, for each , we have where . Thus is a Sorgenfrey open set. As a result, the union of however many is also a Sorgenfrey open set.
Another dramatic difference is the notion of continuous functions. Consider the function in Example above. Even though is not continuous at with respect to the usual open sets, it is continuous with respect to the Sorgenfrey open sets. Note that for the Sorgenfrey open intervals of , only points on the right side of are relevant. Thus, continuous functions with repsect to the Sorgenfrey open sets are called right continuous functions (or upper continuous). A related observation is that with respect to the usual open sets, we can have sequences of points converge to from either side. With respect to Sorgenfrey open sets, we can only have points converge to from the right.
Another difference is that the usual open sets are generated by a metric (a function indicating the distance between any two points). The metric in question is the Euclidean metric ( being the distance between and ). On the other hand, the Sorgenfrey open sets cannot be generated by the Euclidean metric or any other metric. This is a deeper result (see another post A Note On The Sorgenfrey Line in this blog).
We have just presented two examples of topological spaces, both defined on the space of the real numbers. In general, a topological space is a pair where is the underlying space and is the collection of all open sets, which is called a topology. In our first example, and is the set of all open sets generated by the open intervals , which is called the usual topology (or the Euclidean topology of the real line). In the second example, and is the set of all open sets generated by the open intervals , which is called the Sorgenfrey topology. The real line with the Sorgenfrey topology is called the Sorgenfrey line.
Any topological space satisfies the same three conditions stated in Theorem . These three conditions are usually part of the definition of a topological space. So we restate these in the following definition.
A topological space is a pair such that is a set of points and is a collection of subsets of satisfying the following three conditions:
- Both and the empty set belong to ,
- Any union of elements of is also an element of ,
- The intersection of finitely many elements of is also an element of .
In many instances, topological spaces are defined by decalring what the “open intervals” are. Then the open sets are simply the unions of open intervals. The set of all “open intervals” is called a base for a topology. Let be a set. Let be a set of subsets of . If satisfies the following two conditions, is a base for a topology on .
- The set is a cover of (i.e. every point of belongs to some element in ).
- If and , then there is some such that and .
The set of all usual open intervals forms a base for the Euclidean topology of the real line. The set of all open intervals forms a base for the Sorgenfrey topology on the real line. The topology text of Willard is an excellent reference (see ).
- Edwards, C. H. and Penny, D. E., Calculus with Analytic Geometry, Fifth Edition, 1998, Prentice Hall, Upper Saddle River, New Jersey.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.