Consider the real number line with a topology stronger than the Euclidean topology such that the irrational numbers are isolated and the rational numbers retain their Euclidean open neighborhoods. When the real number line is endowed with this topology, the resulting topological space is called the Michael line and is denoted by . It is well known that is not normal where is the space of irrational numbers with the Euclidean topology. This and other basic results about the Michael line are discussed in the post Michael Line Basics. In this post, we show that is paracompact for any positive integer and that (the product of countably and infinitely many copies of ) is not normal. Thus the Michael line is an example demonstrating that even when paracompactness is preserved by taking finite products, it can be destroyed by taking infinite product.
The results discussed in this post are from a paper by E. Michael (Example 1.1 in [2]). This paper had been discussed previously in this blog (see Two footnotes in a paper of E. Michael).
As discussed before, 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 the topology for the Michael line .
Other basic results about the Michael line are discussed in Michael Line Basics.
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Paracompactness
A space is paracompact if every open cover of has a locally finite open refinement. In proving is paracompact, we need two basic results about paracompactness. The proof of Theorem 1 can be found in [1] (Theorem 5.1.11 in page 302) or in [3] (Theorem 20.7 in page 146). We prove Theorem 2.
Theorem 1
Let be a regular space. Then is paracompact if and only if every open cover of has an open locally finite refinement, i.e., the following holds:

every open cover of has an open refinement such that each is a locally finite collection of open subsets of .
Theorem 2
Let be a regular space. Then is paracompact if and only if the following hold:

For each open cover of , there exists a locally finite open cover such that for each .
Proof of Theorem 2
The direction is clear.
Let be paracompact. Let be an open cover of . By regularity, there is an open cover of such that refines . Since is paracompact, has an open locally finite refinement .
We now tie to the original open cover . For each , choose such that . Now, we go the opposite direction, i.e., for each , consider all such that . For each , let be defined by:
Each is open since it is a union of open sets. Since is locally finite, any subcollection of is closure preserving. We have:
Thus we have for all . Since is locally finite, is locally finite. Furthermore, is clearly a cover of . Thus Theorem 2 is established.
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Finite Products
What makes the Michael line finitely productive for paracompactness is that all but countably many points in are isolated. The paracompactness of the finite products of the Michael line follows from Theorem 4 (see Corollary 5 below). Lemma 3 is used in proving Theorem 4.
Lemma 3
Let be a space such that all but countably many points of are isolated. Let be the set of all isolated points of . Then for each , can be expressed as the following:

satisfying the following:
 For each , is homeomorphic to .
 For each , there exists a continuous map such that is the indentity map.
Proof of Lemma 3
Note that is countable. Fix . For each , express . For each and for each , let (the coordinate is fixed and the other coordinates are free to vary). There are only countably many such . Clearly is the union of all . Furthermore, each is homeomorphic to .
Define by mapping each to . In other words, the coordinate of each point is mapped to the fixed point . This is a continuous map since it is a projection map. It is clear that when this map is restricted to , it is the identity map.
When we order all in a sequence , the lemma is established.
Theorem 4
Let be a regular space such that all but countably many points of are isolated. Then is paracompact for each .
Proof of Theorem 4
We prove is paracompact by induction on . Let be the set of all isolated points of . Let .
First we show is paracompact. Let be an open cover of . Enumerate by . For each , choose with . Let . Let be the set of all where . Then is an open locally finite refinement of . By Theorem 1, is paracompact.
Suppose that is paracompact where . Let be an open cover of . By Lemma 3, there exist , all subspaces of , such that:

satisfying the following:
 For each , is homeomorphic to .
 For each , there exists a continuous map such that is the indentity map.
Fix where . Note that is an open cover of . Since each is paracompact, using Theorem 2, we can find a locally finite open refinement (open in ) of such that for each . For each , let .
Then is a locally finite collection of open subsets of covering . Since the map is identity on , . Thus is a cover of . To see that it is locally finite, let . We have . There exists (open in ) such that and only meets finitely many , say, . Consider the following the open sets:
is an open set containing . It follows that the open sets that can meet are limited to ones listed above. For any where , . Thus and . Thus is locally finite in .
For each , let be , which is an open locally finite collection covering (as shown above). All together is an open locally finite collection covering . Let be the set of all where . Then is an open locally finite refinement of . By Theorem 1, is paracompact.
Corollary 5
For each , is paracompact.
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Infinite Products
Let be the space of the irrational numbers with the Euclidean topology. Let be the set of all nonnegative integers. We now show that , the product of countably and infinitely many copies of the Michael line, is not normal. Before doing that, we point out that when is considered a discrete space, , the product of countably and infinitely many copies of , is homeomorphic to (Thinking about the Space of Irrationals Topologically).
Let be a countably infinite subset of the Michael line such that is closed and discrete. As discussed above, is a homeomorphic copy of . Furthermore is a closed subset of . Thus contains as a closed subspace. Since is not normal, is not normal. On the other hand, is homeomorphic to . Thus is not normal.
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Reference
 Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
 Michael, E., Paracompactness and the Lindelof property in Finite and Countable Cartesian Products, Compositio Math., 23, 1971, 199214.
 Willard, S., General Topology, 1970, AddisonWesley Publishing Company.
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