Like the Sorgenfrey line, the Michael line is a classic counterexample that is covered in standard topology textbooks and in first year topology courses. This easily accessible example helps transition students from the familiar setting of the Euclidean topology on the real line to more abstract topological spaces. One of the most famous results regarding the Michael line is that the product of the Michael line with the space of the irrational numbers is not normal. Thus it is an important example in demonstrating the pathology in products of paracompact spaces. The product of two paracompact spaces does not even have be to be normal, even when one of the factors is a complete metric space. In this post, we discuss this classical result and various other basic results of the Michael line.
Let be the real number line. Let be the set of all irrational numbers. Let , the set of all rational numbers. Let be the usual topology of the real line . The following is a base that defines a topology on .
The real line with the topology generated by is called the Michael line and is denoted by . In essense, in , points in are made isolated and points in retain the usual Euclidean open sets.
The Euclidean topology is coarser (weaker) than the Michael line topology (i.e. being a subset of the Michael line topology). Thus the Michael line is Hausdorff. Since the Michael line topology contains a metrizable topology, is submetrizable (submetrized by the Euclidean topology). It is clear that is first countable. Having uncountably many isolated points, the Michael line does not have the countable chain condition (thus is not separable). The following points are discussed in more details.
The product is not normal.
There exist a Lindelof space and a separable metric space such that is not normal.
- The space is paracompact.
- The space is not Lindelof.
- The extent of the space is where is the cardinality of the real line.
- The space is not locally compact.
- The space is not perfectly normal, thus not metrizable.
- The space is not a Moore space, but has a -diagonal.
- The product is not normal where has the usual topology.
- The product is metacompact.
- The space has a point-countable base.
- For each , the product is paracompact.
Results 10, 11 and 12 are shown in some subsequent posts.
Baire Category Theorem
Before discussing the Michael line in greater details, we point out one connection between the Michael line topology and the Euclidean topology on the real line. The Michael line topology on coincides with the Euclidean topology on . A set is said to be a -set if it is the intersection of countably many open sets. By the Baire category theorem, the set is not a -set in the Euclidean real line (see the section called “Discussion of the Above Question” in the post A Question About The Rational Numbers). Thus the set is not a -set in the Michael line. This fact is used in Result 5.
The fact that is not a -set in the Euclidean real line implies that is not an -set in the Euclidean real line. This fact is used in Result 7.
Let be an open cover of . We proceed to derive a locally finite open refinement of . Recall that is the usual topology on . Assume that consists of open sets in the base . Let . Let . Note that is a Euclidean open subspace of the real line (hence it is paracompact). Then there is such that is a locally finite open refinement of and such that covers (locally finite in the Euclidean sense). Then add to all singleton sets where and let denote the resulting open collection.
The resulting is a locally finite open collection in the Michael line . Furthermore, is also a refinement of the original open cover .
A similar argument shows that is hereditarily paracompact.
To see that is not Lindelof, observe that there exist Euclidean uncountable closed sets consisting entirely of irrational numbers (i.e. points in ). For example, it is possible to construct a Cantor set entirely within .
Let be an uncountable Euclidean closed set consisting entirely of irrational numbers. Then this set is an uncountable closed and discrete set in . In any Lindelof space, there exists no uncountable closed and discrete subset. Thus the Michael line cannot be Lindelof.
The argument in Result 2 indicates a more general result. First, a brief discussion of the cardinal function extent. The extent of a space is the smallest infinite cardinal number such that every closed and discrete set in has cardinality . The extent of the space is denoted by . When the cardinal number is (the first infinite cardinal number), the space is said to have countable extent, meaning that in this space any closed and discrete set must be countably infinite or finite. When , there are uncountable closed and discrete subsets in the space.
It is straightforward to see that if a space is Lindelof, the extent is . However, the converse is not true.
The argument in Result 2 exhibits a closed and discrete subset of of cardinality . Thus we have .
The Michael line is not locally compact at all rational numbers. Observe that the Michael line closure of any Euclidean open interval is not compact in .
A set is said to be a -set if it is the intersection of countably many open sets. A space is perfectly normal if it is a normal space with the additional property that every closed set is a -set. In the Michael line , the set of rational numbers is a closed set. Yet, is not a -set in the Michael line (see the discussion above on the Baire category theorem). Thus is not perfectly normal and hence not a metrizable space.
The diagonal of a space is the subset of its square that is defined by . If the space is Hausdorff, the diagonal is always a closed set in the square. If is a -set in , the space is said to have a -diagonal. It is well known that any metric space has -diagonal. Since is submetrizable (submetrized by the usual topology of the real line), it has a -diagonal too.
Any Moore space has a -diagonal. However, the Michael line is an example of a space with -diagonal but is not a Moore space. Paracompact Moore spaces are metrizable. Thus is not a Moore space. For a more detailed discussion about Moore spaces, see Sorgenfrey Line is not a Moore Space.
We now show that is not normal where has the usual topology. In this proof, the following two facts are crucial:
- The set is not an -set in the real line.
- The set is dense in the real line.
Let and be defined by the following:
The sets and are disjoint closed sets in . We show that they cannot be separated by disjoint open sets. To this end, let and where and are open sets in .
To make the notation easier, for the remainder of the proof of Result 7, by an open interval , we mean the set of all real numbers with . By , we mean . For each , choose an open interval such that . We also assume that is the midpoint of the open interval . For each positive integer , let be defined by:
Note that . For each , let (Euclidean closure in the real line). It is clear that . On the other hand, (otherwise would be an -set in the real line). So there exists such that . So choose a rational number such that .
Choose a positive integer such that . Since is dense in the real line, choose such that . Now we have . Choose another integer such that and .
Since , choose such that . Now it is clear that . The following inequalities show that .
The open interval is chosen to have length . Since , . Thus . We have shown that . Thus is not normal.
As indicated above, the proof of Result 7 hinges on two facts about , namely that it is not an -set in the real line and it is dense in the real line. We can modify the construction of the Michael line by using other partition of the real line (where one set is isolated and its complement retains the usual topology). As long as the set that is isolated is not an -set in the real line and is dense in the real line, the same proof will show that the product of the modified Michael line and the space (with the usual topology) is not normal. This will be how Result 12 is derived.
The product is not paracompact since it is not normal. However, is metacompact.
A collection of subsets of a space is said to be point-finite if every point of belongs to only finitely many sets in the collection. A space is said to be metacompact if each open cover of has an open refinement that is a point-finite collection.
Note that . The first in is discrete (a subspace of the Michael line) and the second has the Euclidean topology.
Let be an open cover of . For each , choose such that . We can assume that where is a usual open interval in and is a usual open interval in . Let .
Fix . For each , choose some such that . We can assume that where is a usual open interval in . Let .
As a subspace of the Euclidean plane, is metacompact. So there is a point-finite open refinement of . For each , has a point-finite open refinement . Let be the union of and all the where . Then is a point-finite open refinement of .
Note that the point-finite open refinement may not be locally finite. The vertical open intervals in , can “converge” to a point in . Thus, metacompactness is the best we can hope for.
A collection of sets is said to be point-countable if every point in the space belongs to at most countably many sets in the collection. A base for a space is said to be a point-countable base if , in addition to being a base for the space , is also a point-countable collection of sets. The Michael line is an example of a space that has a point-countable base and that is not metrizable. The following is a point-countable base for :
where is the set of all Euclidean open intervals with rational endpoints. One reason for the interest in point-countable base is that any countable compact space (hence any compact space) with a point-countable base is metrizable (see Metrization Theorems for Compact Spaces).
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.