Monthly Archives: January 2018

Drawing more Sorgenfrey continuous functions

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In this previous post, we draw continuous functions on the Sorgenfrey line S to gain insight about the function C_p(S). In this post, we draw more continuous functions with the goal of connecting C_p(S) and C_p(D) where D is the double arrow space. For example, C_p(D) can be embedded as a subspace of C_p(S). More interestingly, both function spaces C_p(D) and C_p(S) share the same closed and discrete subspace of cardinality continuum. As a result, the function space C_p(D) is not normal.

Double Arrow Space

The underlying set for the double arrow space is D=[0,1] \times \{ 0,1 \}, which is a subset in the Euclidean plane.

Figure 1 – The Double Arrow Space

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.

Figure 2 – Open Neighborhoods in the Double Arrow Space

More specifically, for any a with 0 \le a < 1, a basic open set containing the point (a,1) is of the form \displaystyle \biggl[ [a,b) \times \{ 1 \} \biggr] \cup \biggl[ (a,b) \times \{ 0 \} \biggr], painted red in Figure 2. One the other hand, for any a with 0<a \le 1, a basic open set containing the point (a,0) is of the form \biggl[ (c,a) \times \{ 1 \} \biggr] \cup \biggl[ (c,a] \times \{ 0 \} \biggr], painted blue in Figure 2. The upper right point (1,1) and the lower left point (0,0) 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 C_p(D) can be embedded as a subspace in the function C_p(S).
  • Both function spaces C_p(D) and C_p(S) share the same closed and discrete subspace of cardinality continuum.
  • The function space C_p(D) is not normal.

Drawing a Map from Sorgenfrey Line onto Double Arrow Space

In order to show that C_p(D) can be embedded into C_p(S), we draw a continuous map from the Sorgenfrey line S onto the double arrow space D. The following diagram gives the essential idea of the mapping we need.

Figure 3 – Mapping Sorgenfrey Line onto Double Arrow Space

The mapping shown in Figure 3 is to map the interval [0,1] onto the upper line segment of the double arrow space, as demonstrated by the red arrow. Thus x \mapsto (x,1) for any x with 0 \le x \le 1. Essentially on the interval [0,1], the mapping is the identity map.

On the other hand, the mapping is to map the interval [-1,0) onto the lower line segment of the double arrow space less the point (0, 0), as demonstrated by the blue arrow in Figure 3. Thus -x \mapsto (x,0) for any -x with 0<x \le 1. Essentially on the interval [-1,0), the mapping is the identity map times -1.

The mapping described by Figure 3 only covers the interval [-1,1] in the domain. To complete the mapping, let x \mapsto (1,1) for any x \in (1, \infty) and x \mapsto (0,0) for any x \in (-\infty, -1).

Let h 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 h: S \rightarrow D is continuous.

Embedding

We use the following fact to show that C_p(D) can be embedded into C_p(S).

Suppose that the space Y is a continuous image of the space X. Then C_p(Y) can be embedded into C_p(X).

Based on this result, C_p(D) can be embedded into C_p(S). The embedding that makes this true is E(f)=f \circ h for each f \in C_p(D). Thus each function f in C_p(D) is identified with the composition f \circ h where h is the map defined in Figure 3. The fact that E(f) 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 C_p(S).

Figure 4 – a family of Sorgenfrey continuous functions

For each 0<a<1, let f_a: S \rightarrow \{0,1 \} be the continuous function described in Figure 4. The previous post shows that the set F=\{ f_a: 0<a<1 \} is a closed and discrete subspace of C_p(S). We claim that F \subset C_p(D) \subset C_p(S).

To see that F \subset C_p(D), we define continuous functions U_a: D \rightarrow \{0,1 \} such that f_a=U_a \circ h. We can actually back out the map U_a from f_a in Figure 4 and the mapping h. Here’s how. The function f_a is piecewise constant (0 or 1). Let’s focus on the interval [-1,1] in the domain of f_a.

Consider where the function f_a maps to the value 1. There are two intervals, [a,1) and [-1,-a), where f_a maps to 1. The mapping h maps [a,1) to the set [a,1) \times \{ 1 \}. So the function U_a must map [a,1) \times \{ 1 \} to the value 1. The mapping h maps [-1,-a) to the set (a,1] \times \{ 0 \}. So U_a must map (a,1] \times \{ 0 \} to the value 1.

Now consider where the function f_a maps to the value 0. There are two intervals, [0,a) and [-a,0), where f_a maps to 0. The mapping h maps [0,a) to the set [0,a) \times \{ 1 \}. So the function U_a must map [0,a) \times \{ 1 \} to the value 0. The mapping h maps [-a,0) to the set (0,a] \times \{ 0 \}. So U_a must map (0,a] \times \{ 0 \} to the value 0.

To take care of the two isolated points (1,1) and (0,0) of the double arrow space, make sure that U_a maps these two points to the value 0. The following is a precise definition of the function U_a.

    \displaystyle U_a(y) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ y \in [a,1) \times \{ 1 \} \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ y \in (a,1] \times \{ 0 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y \in (0,a] \times \{ 0 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y \in [0,a) \times \{ 1 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y=(0,0) \text{ or } y = (1,1) \end{array} \right.

The resulting U_a is a translation of f_a. Under the embedding E defined earlier, we see that E(U_a)=f_a. Let U=\{ U_a: 0<a<1 \}. The set U in C_p(D) is homeomorphic to the set F in C_p(S). Thus U is a closed and discrete subspace of C_p(D) since F is a closed and discrete subspace of C_p(S).

Remarks

The drawings and the embedding discussed here and in the previous post establish that C_p(D), the space of continuous functions on the double arrow space, contains a closed and discrete subspace of cardinality continuum. It follows that C_p(D) is not normal. This is due to the fact that if C_p(X) is normal, then C_p(X) must have countable extent (i.e. all closed and discrete subspaces must be countable).

While C_p(D) is embedded in C_p(S), the function space C_p(S) is not embedded in C_p(D). Because the double arrow space is compact, C_p(D) has countable tightness. If C_p(S) were to be embedded in C_p(D), then C_p(S) would be countably tight too. However, C_p(S) is not countably tight due to the fact that S \times S is not Lindelof (see Theorem 1 in this previous post).

Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Tkachuk V. V., A C_p-Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.

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