In this previous post, we draw continuous functions on the Sorgenfrey line to gain insight about the function . In this post, we draw more continuous functions with the goal of connecting and where is the double arrow space. For example, can be embedded as a subspace of . More interestingly, both function spaces and share the same closed and discrete subspace of cardinality continuum. As a result, the function space is not normal.
Double Arrow Space
The underlying set for the double arrow space is , which is a subset in the Euclidean plane.
The name of double arrow comes from the fact that an open neighborhood of a point in the upper line segment points to the right while an open neighborhood of a point in the lower line segment points to the left. This is demonstrated in the following diagram.
More specifically, for any with , a basic open set containing the point is of the form , painted red in Figure 2. One the other hand, for any with , a basic open set containing the point is of the form , painted blue in Figure 2. The upper right point and the lower left point are made isolated points.
The double arrow space is a compact space that is perfectly normal and not metrizable. Basic properties of this space, along with those of the lexicographical ordered space, are discussed in this previous post.
The drawing of continuous functions in this post aims to show the following results.
- The function space can be embedded as a subspace in the function .
- Both function spaces and share the same closed and discrete subspace of cardinality continuum.
- The function space is not normal.
Drawing a Map from Sorgenfrey Line onto Double Arrow Space
In order to show that can be embedded into , we draw a continuous map from the Sorgenfrey line onto the double arrow space . The following diagram gives the essential idea of the mapping we need.
The mapping shown in Figure 3 is to map the interval onto the upper line segment of the double arrow space, as demonstrated by the red arrow. Thus for any with . Essentially on the interval , the mapping is the identity map.
On the other hand, the mapping is to map the interval onto the lower line segment of the double arrow space less the point , as demonstrated by the blue arrow in Figure 3. Thus for any with . Essentially on the interval , the mapping is the identity map times -1.
The mapping described by Figure 3 only covers the interval in the domain. To complete the mapping, let for any and for any .
Let be the mapping that has been described. It maps the Sorgenfrey line onto the double arrow space. It is straightforward to verify that the map is continuous.
We use the following fact to show that can be embedded into .
Based on this result, can be embedded into . The embedding that makes this true is for each . Thus each function in is identified with the composition where is the map defined in Figure 3. The fact that is an embedding is shown in this previous post (see Theorem 1).
Same Closed and Discrete Subspace in Both Function Spaces
The following diagram describes a closed and discrete subspace of .
For each , let be the continuous function described in Figure 4. The previous post shows that the set is a closed and discrete subspace of . We claim that .
To see that , we define continuous functions such that . We can actually back out the map from in Figure 4 and the mapping . Here’s how. The function is piecewise constant (0 or 1). Let’s focus on the interval in the domain of .
Consider where the function maps to the value 1. There are two intervals, and , where maps to 1. The mapping maps to the set . So the function must map to the value 1. The mapping maps to the set . So must map to the value 1.
Now consider where the function maps to the value 0. There are two intervals, and , where maps to 0. The mapping maps to the set . So the function must map to the value 0. The mapping maps to the set . So must map to the value 0.
To take care of the two isolated points and of the double arrow space, make sure that maps these two points to the value 0. The following is a precise definition of the function .
The resulting is a translation of . Under the embedding defined earlier, we see that . Let . The set in is homeomorphic to the set in . Thus is a closed and discrete subspace of since is a closed and discrete subspace of .
The drawings and the embedding discussed here and in the previous post establish that , the space of continuous functions on the double arrow space, contains a closed and discrete subspace of cardinality continuum. It follows that is not normal. This is due to the fact that if is normal, then must have countable extent (i.e. all closed and discrete subspaces must be countable).
While is embedded in , the function space is not embedded in . Because the double arrow space is compact, has countable tightness. If were to be embedded in , then would be countably tight too. However, is not countably tight due to the fact that is not Lindelof (see Theorem 1 in this previous post).
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Tkachuk V. V., A -Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.
Dan Ma math
Daniel Ma mathematics