The Euclidean topology of the real line (1)

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 \mathbb{R} be the set of all real numbers. For any two real numbers a and b with a<b, by the open interval (a,b), we mean the set of all real numbers x such that a<x<b. Let U be a subset of \mathbb{R}. The set U is said to be an open set if for each x \in U, there is an open interval (a,b) containing x such that (a,b) \subset U. A set C \subset \mathbb{R} is said to be a closed set if the complement \mathbb{R}-C is an open set.

By definition, all open intervals (a,b) are open sets. The set (0,1) \cup (2,3) is an open set. Note that it is the union of two open intervals. The set of all positive numbers (0,\infty) is open. Note that (0,\infty)=\bigcup \limits_{x>0}(0,x).

Let \tau be the set of all subsets of \mathbb{R} that are open sets. The set \tau 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 1).

Theorem 1

  1. Both \mathbb{R} and the empty set \phi are open sets.
  2. The union of open sets is an open set.
  3. 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: \mathbb{N}, \mathbb{Q}, \mathbb{P}, [0,1]. 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 Y \subset \mathbb{R}, Y can also have a topology. The open sets in the subspace Y are inherited from the overall real line (i.e. they are simply the open sets in the overall real line but points not in Y are excluded). For example, for each nonnegative integer x, the singleton set \lbrace{x}\rbrace is obviously not an open set in the real line. However \lbrace{x}\rbrace, where x \in \mathbb{N}, is an open set in the space \mathbb{N}. This is because we have:

\mathbb{N} \cap (x-0.1,x+0.1)=\lbrace{x}\rbrace

Since every single point in \mathbb{N} is an open set, the space \mathbb{N} is said to be a discrete space. An open interval in \mathbb{Q} is simply (a,b) \cap \mathbb{Q} (i.e. the set of all rational bumbers x with a<x<b). Note that every open interval contains infinitely many rational numbers. Thus \mathbb{Q} is not a discrete space. For the unit interval [0,1], an open interval containing the point 1 is (a,1]=[0,1] \cap (a,b) where 0<a<1<b. Likewise, an open interval containing the left endpoint 0 is [0,c). In general, for the subset Y \subset \mathbb{R}, the open sets are generated by the open intervals of the form (a,b) \cap Y. The set of all open sets just defined for the subset Y is called the relative topology for Y 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 \epsilon \delta-defintion is used. We consider continuous functions from a topological point of view.

Defintion 1
Let’s see how continuous function is defined in a typical calculus text such as [1]. The function f: (s,t) \rightarrow \mathbb{R} is continuous at a \in (s,t) if \lim \limits_{x \rightarrow a} f(x)=f(a). For the definition of \lim \limits_{x \rightarrow a} F(x)=L, we have the following from [1]:

The number L is the limit of F(x) as x approaches a provided that, given any number \epsilon>0, there exists a number \delta>0 such that \lvert F(x)-L \lvert<\epsilon for all x such that 0<\lvert x-a \lvert <\delta.

Defintion 2
Let X \subset \mathbb{R} and let f: X \rightarrow \mathbb{R} be a function. The function f is said to be continuous at x \in X if for each open interval (a,b) containing f(x), there is some open interval (c,d) containing x such that f(X \cap (c,d)) \subset (a,b). The function f is said to be a continuous function if it is continuous at every x \in X.

Of course, definition 2 is equivalent to defintion 1. In definition 2, if f(x) is the midpoint of the open interval (a,b), then \epsilon=0.5(b-a) corresponds to the \epsilon in definition 1. Likewise \delta=0.5(d-c) is the \delta in definition 1.

Defintion 1 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 1 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 2, 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 2. We have the following characterizations of continuous functions:

Theorem 2
Let X \subset \mathbb{R} and let f: X \rightarrow \mathbb{R} be a function. Then the following conditions are equivalent:

  1. The function f is continuous.
  2. For each open set U \subset \mathbb{R}, the inverse image f^{-1}(U) is an open set in the domain space X.
  3. For each closed set C \subset \mathbb{R}, the inverse image f^{-1}(C) is a closed set in the domain space X.

Example 1
Some familiar examples of continuous functions include polynomial functions with real coefficients, trigonometric functions such as sin(x) and cos(x) as well as exponential functions such as e^x and logarithmic functions such as ln(x).

Example 2
The following function F(x) is not continuous at x=0.

\displaystyle F(x)=\left\{\begin{matrix}0&\thinspace x<0\\{1}&\thinspace x\ge0\end{matrix}\right.

Example 3
The following function G(x) is not continuous at every x \in \mathbb{R}.

\displaystyle G(x)=\left\{\begin{matrix}1&\thinspace \text{x is a rational number}\\{0}&\thinspace \text{x is an irrational number}\end{matrix}\right.

Example 4
Let A=\lbrace{x \in \mathbb{R}: x \ne 0}\rbrace. Define H(x):A \rightarrow \mathbb{R} by H(x)=\frac{1}{x}. The function H(x) is continuous at every x \in A. Thus it is a continuous function (as a function defined over A). 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 [a,b) where a<b. In other words, the open intervals include the left endpoint. As before, a set U \subset \mathbb{R} is open if for each point x \in U, there is an open interval [a,b) containing x such that [a,b) \subset U. Theorem 1 still hold true with this definition of open sets defined in the real line. Let \tau_s be the set of all open sets generated by the open intervals [a,b). The set \tau_s is called the Sorgenfrey topology of the real line. Thus we have two topologies defined on \mathbb{R}, \tau and \tau_s. To avoid confusion, when we discuss the both types of open sets at the same time, we refer to the open sets in \tau (defined by the open intervals (a,b)) as the Euclidean open sets or the usual open sets. We refer to the open sets in \tau_s (defined by the open intervals [a,b)) 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. \tau \subset \tau_s). To see this, for each x \in (a,b), we have [x,t) \subset (a,b) where x<t<b. Thus (a,b) is a Sorgenfrey open set. As a result, the union of however many (a,b) is also a Sorgenfrey open set.

Another dramatic difference is the notion of continuous functions. Consider the function F(x) in Example 2 above. Even though F(x) is not continuous at x=0 with respect to the usual open sets, it is continuous with respect to the Sorgenfrey open sets. Note that for the Sorgenfrey open intervals [0,c) of x=0, only points on the right side of x=0 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 x \in \mathbb{R} from either side. With respect to Sorgenfrey open sets, we can only have points converge to x \in \mathbb{R} 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 (\lvert x-y \lvert being the distance between x and y). 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).

Topological Spaces
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 (X, \tau) where X is the underlying space and \tau_0 is the collection of all open sets, which is called a topology. In our first example, X=\mathbb{R} and \tau_0=\tau is the set of all open sets generated by the open intervals (a,b), which is called the usual topology (or the Euclidean topology of the real line). In the second example, X=\mathbb{R} and \tau_0=\tau_s is the set of all open sets generated by the open intervals [a,b), which is called the Sorgenfrey topology. The real line with the Sorgenfrey topology is called the Sorgenfrey line.

Any topological space (X, \tau_0) satisfies the same three conditions stated in Theorem 1. These three conditions are usually part of the definition of a topological space. So we restate these in the following definition.

Definition 3
A topological space is a pair (X, \tau_0) such that X is a set of points and \tau_0 is a collection of subsets of X satisfying the following three conditions:

  1. Both X and the empty set \phi belong to \tau_0,
  2. Any union of elements of \tau is also an element of \tau_0,
  3. The intersection of finitely many elements of \tau_0 is also an element of \tau_0.

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 X be a set. Let \mathcal{B} be a set of subsets of X. If \mathcal{B} satisfies the following two conditions, \mathcal{B} is a base for a topology on X.

  • The set \mathcal{B} is a cover of X (i.e. every point of X belongs to some element in \mathcal{B}).
  • If B_1,B_2 \in \mathcal{B} and x \in B_1 \cap B_2, then there is some B_3 \in \mathcal{B} such that x \in B_3 and B_3 \subset B_1 \cap B_2.

The set of all usual open intervals (a,b) forms a base for the Euclidean topology of the real line. The set of all open intervals [a,b) forms a base for the Sorgenfrey topology on the real line. The topology text of Willard is an excellent reference (see [2]).

Reference

  1. Edwards, C. H. and Penny, D. E., Calculus with Analytic Geometry, Fifth Edition, 1998, Prentice Hall, Upper Saddle River, New Jersey.
  2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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3 thoughts on “The Euclidean topology of the real line (1)

  1. Pingback: Topology [1] : An introduction to Point Set Topology | fundamentally trivial

  2. If u can give us an example to explain why the closure of union of infinite set in this topology not equal to the union of closure of infinite set

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