# Drawing more Sorgenfrey continuous functions

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 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. 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, 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 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. 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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Dan Ma math

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