This post is a basic discussion on the unit square with the lexicographic order and one of its subspace called the double arrow space. The goal is to establish several basic facts of these two spaces. Both of these spaces are compact non-metrizable spaces. The double arrow space consists of the top and bottom edges of the unit square. The double arrow has a strong connection to the Sorgenfrey Line and its square is an example of a space that is normal that is not completely normal. The unit square with the lexicographic order is an example of a completely normal space that is not perfectly normal.
Defining the spaces
Let be the unit interval . Let be . Consider the lexicographic order on . This is the linear order defined by letting whenever or and . The goal is to consider the square with the topology induced by this linear order. For and , is the notation in this post for open intervals in the unit square.
To facilitate further discussion, let’s look at the following subsets of .
The subspace is called the double arrow space. In this post, my aim is to establish some basic facts about the double arrow space and then the unit square .
Double arrow space
A. is compact.
B. is hereditarily Lindelof and hereditarily separable.
C. is perfectly normal.
D. is not hereditarily normal.
Unit square with lexicographic order
E. is compact.
F. is first countable.
G. is not separable.
H. does not have the countable chain condition (ccc).
I. is not perfectly normal.
Open intervals in these spaces
Before proving the results stated above, let’s make sense of the open intervals in and then in . The open intervals for points in are essentially the usual open intervals. For example, for the point , the following is an open interval in .
Because of this, has an uncountable collection of pairwise disjoint open intervals (thus does not have ccc and is not separable). Also, a vertical line in of the form is a homeomorphic copy of the unit interval with the usual topology.
The more interesting open neighborhoods are for the points in the top and bottom lines (i.e. the double arrow space). Let’s look an example of an open interval in for the point . Consider the open interval where and . It follows that the open interval where
Note that is the vertical strip in the middle, is the left edge, and is the right edge. If you look at , then the open interval for the point becomes
The above is also an open interval in containing the point . Similarly, an example of an open interval containing the point is:
Based on the above examples of open intervals, open intervals in have the following form, hence the term “double arrow”.
The double arrow space contains two copies of the Sorgenfrey line. So knowing the basic properties of the Sorgenfrey line will help follow the proofs of A through D below. Information about the Sorgenfrey line can be found in the following blog post.
Proof of A. I now proof that the double arrow space is compact. I do so by showing that it is Lindelof and countably compact. A space is countably compact if any countable open cover has a finite subcover (equivalently, any infinite subspace has a limit point). Here, limit point is in the topological sense (i.e. the point is a limit point of the set if any open set containing contains a point of different from ).
Note that and as defined above are homeomorphic copies of the Sorgenfrey Line. Thus is made of of two copies of the Sorgenfrey Line plus 4 points. This shows that is hereditarily Lindelof.
Pick an infinite subset of . Then either the top or the bottom contains infinitely many points of . Assume the top does. There is a point that is a limit point of in the usual topology. If the point is a right-sided limit point of in the usual topology, then the point is a limit point of in the order topology. If the point is a left-sided limit point of in the usual topology, then the point is a limit point of in the order topology.
Proof of B, C and D. As observed in the proof of A, the double arrow space is the union of 2 copies of the Sorgenfrey Line plus 4 points. Thus it is hereditarily Lindelof, hereditarily separable and perfectly normal.
Note that contains a copy of the Sorgenfrey Plane. Thus it is not hereditarily normal. This also implies that both the double arrow space and the unit square with the lexicographic order are not metrizable.
Proof of E. Let be an open cover for that consists of open intervals. Since is compact, there exists a finite collection of open intervals from that cover . Based on the above observation about open intervals of points in the double arrow space, the open intervals cover all the points in except possibly the left and right edges of the open intervals . With the topology inherited from the order topology, each of the edges is homeomorphic to the unit interval with the usual topology. Thus there are finitely many open intervals from that cover these left and right edges. Thus has a finite subcover and is compact.
Proof of F, G, H. These are clear based on the above observation made about the open intervals in the unit square.
Proof of I. I show here that cannot be a set in the unit square . For each , let be an open set in such that . Since is compact, we can assume that is the union of finitely many open intervals. Based on the observation given above, each covers all of except for finitely many vertical lines (the left and right edges of these open intervals). Pick one vertical line that is not one of the vertical edges from the open intervals . Clearly this vertical line is covered by for each . Thus is not a set in .
Comment. The double arrow space is made up of 2 copies of the Sorgenfrey Line and the 4 corner points. It follows that it is hereditarily Lindelof and perfectly normal. However, since the square of the double arrow space contains a copy of the Sorgenfrey Plane, the square of the double arrow space is not hereditarily normal, thus showing that normality is not a hereditary notion. The square of the double arrow space is a handy example of a normal space that is not completely normal. This implies both the double arrow space and the unit square with the lexicographic order not metrizable. The space also demonstrates that hereditarily normality is not preserved by Cartesian product.
The unit square with the lexicographic order topology is completely normal ([Steen & Seebach]). Thus it is an example of a space that completely normal (T4) but not perfectly normal (T5).
[Steen & Seebach]
Steen, L. A. and Seebach, J. A.,  Counterexamples In Topology, Dover Edition