This is another discussion of the Euclidean topology of the real line. The previous post (The Euclidean topology of the real line (1)) introduced the notion of topological spaces via a discussion of the Euclidean (usual) topology of the real number line. In this post we continue to discuss the usual topology of the real line, mostly basic facts about open sets and closed sets that we will need in subsequent posts. Also, much of what we discuss in this post about the usual topology will hold for topological spaces in general. This post is part of a series of posts on the topology on the real line. We believe that the Euclidean topology (especially the real line topology and the real plane topology) is a gateway to general topology.
This is a post on introductory topology. Concepts discussed here are basic notions found in beginning topology courses and elementary analysis courses. For the readers who are taking math courses that transition their math career from calculation to proving of theorems, we encourage such readers to think about the concepts and theorems discussed here before reading the proofs.
Let be the real number line.
- An open interval is the set of all real numbers such that .
- A subset of is an open set if for each , there is some open interval containing such that .
- The complement of (denoted by ) is the set of all points such that .
- A set is a closed set if its complement is an open set.
- A point is said to be a limit point of the set if every open set containing contains a point of different from .
- A set is a perfect set if is closed and every point of is a limit point of .
- For any set , let denote the set of all limit points of . The set is said to be the derived set of .
- The closure of is the set .
- A set is dense in if .
Since an open set of made up of open intervals, we have the following observations about open sets (stated in the following theorem).
- Both and the empty set are open sets.
- Let be a collection of open sets where is some index set. Then the union of the open sets , denoted by , is also an open set.
- If are open sets where is a positive integer, then is an open set.
The three conditions in this theorem are the axiomatic conditions for a collection of open sets to be a topology. Once we show that the open sets as defined in this post satisfy these three conditions, we demonstrate that the set of all open subsets of is a topology, which is the Euclidean topology.
It is clear that is open since for each point , we have . Since the empty set has no point, it satisfies the definition of open set by default.
Let be a collection of open sets where is some index set. Let . Then for some . By definition, there is some open interval such that . Note that .
To show condition , it suffices to show that the intersection of two open sets is open. Let where and are open. Then there are open intervals and such that and . We can find a smaller open interval such that and . Thus we have . This shows that is an open set.
Note that we can set in the proof of condition . In many other topological spaces, the intersection of two “open intervals” is not necessarily an “open interval”. However, we can always find an open interval contained within the intersection of two open intervals.
Since a closed set is the complement of an open set, closed sets satisfy the following conditions:
- Both and the empty set are closed sets.
- Let be a collection of closed sets where is some index set. Then the intersection of the closed sets , denoted by , is also a closed set.
- If are open sets where is a positive integer, then is a closed set.
- If the point is a limit point of the set , then every open set containing containing infinitely many points of .
- Finite sets of real numbers have no limit points.
Let be a limit point of . It suffices to show that every open interval containing contains infinitely many points of . Let be an open interval such that .
Suppose that only contains finitely many points of and we aim to derive a contradiction. Suppose these finite number of points are . Then we take the minimum of the following differences:
All of the above differences are positive as the points are different from . Let be the smallest of the above differences. Consider the open interval . Note that points in this open intervals are less than away from the point . The points are at least away from . Thus we have an open interval containing that has no points of other than , contradicting the fact that is a limit point of . So, the assumption above that there is an open interval containing that contains only finitely many points of is faulty and the opposite must be true.
A corollary to the above is that only infinite set can have a limit point. Thus every finite set of real numbers has no limit point.
Let . Then the following conditions are equivalent.
- The set is a closed set.
- If is a limit point of , then .
Suppose is closed. Suppose we have: is a limit point of and . Since is open and , contains a point of , which is impossible. Thus if is a limit point of , then .
Condition is really just a more compact way of expressing condition . Note that where is the set of all limit points of . Also note that . Another way of saying condition is that .
We prove the contrapositive: if is not closed, then has a limit point that is not in . To see this, note that if is not closed, then is not open. Then there is a point such that for each open interval containing , is not a subset of (this means contains a point of ). In other words, is a limit point of and .
The following proposition easily follows from definition. For emphasis, we would like to call that out in a proposition.
A set is dense in if and only if every nonempty open set of contains points of .
Consider the following sets:
The set is closed. The following is the complement of , which is a union of open sets.
The set has no limit point. Note that for each , is an open interval containing no point of other than . A closed set that has no limit point is said to be a discrete set.
The point is a limit point of . Note that every open interval where contains for some sufficiently large positive integer . None of the point in is a limit point of . We have .
The comments made about the set can be made about the set . The point is a limit point of . We have .
The set is the union of an open interval and a single point. It is not an open set in since no open interval containing the point is a subset of .
The set is dense in . It is a property of the real line that for each pair of real numbers with there is some rational number with . Since is a countable set, the real line has a countable dense set. This is an important property among topological spaces. Though is not closed, every point of is a limit point of .
The unit interval is closed as its complement is the open set . Every point of is a limit point of . Thus is a perfect set. A more interesting point about the unit interval is that it is a closed and bounded set and thus is a compact space. We will discuss this in a subsequent post.
Very often we need to work with a subset of as a topological space. The open sets for a subset are inherited from the overall space . Let . In working with as a topological space, we only consider points of . Thus the open sets for are of the form where is an open set in the real line. We say that is an open set relative to . As the following examples show, a set can be open in the relative topology and not open in the overall topology.
The set in the above example is clearly not an open set in . However, if we consider the space as a space then is an open set. Within the confine of the space , the singleton set is an open set (i.e. open relative to ).
Let be the set of all irrational numbers. Consider as a topological space with open sets inherited from . The set has no limit point in since the point is no longer in the space. It is clear that no point of is a limit point of . Thus is closed set in . Note that is not a }closed set in .
The following are two excellent texts at the introductory level.
- Steen, L. A. and Seebach, J. A., Counterexamples in Topology, 1995, Dover Publications, Inc, New York.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.