Cp(omega 1 + 1) is not normal

In this and subsequent posts, we consider C_p(X) where X is a compact space. Recall that C_p(X) is the space of all continuous real-valued functions defined on X and that it is endowed with the pointwise convergence topology. One of the compact spaces we consider is \omega_1+1, the first compact uncountable ordinal. There are many interesting results about the function space C_p(\omega_1+1). In this post we show that C_p(\omega_1+1) is not normal. An even more interesting fact about C_p(\omega_1+1) is that C_p(\omega_1+1) does not have any dense normal subspace [1].

Let \omega_1 be the first uncountable ordinal, and let \omega_1+1 be the successor ordinal to \omega_1. The set \omega_1 is the first uncountable ordinal. Furthermore consider these ordinals as topological spaces endowed with the order topology. As mentioned above, the space \omega_1+1 is the first compact uncountable ordinal. In proving that C_p(\omega_1+1) is not normal, a theorem that is due to D. P. Baturov is utilized [2]. This theorem is also proved in this previous post.

For the basic working of function spaces with the pointwise convergence topology, see the post called Working with the function space Cp(X).

The fact that C_p(\omega_1+1) is not normal is established by the following two points.

  • If C_p(\omega_1+1) is normal, then C_p(\omega_1+1) has countable extent, i.e. every closed and discrete subspace of C_p(\omega_1+1) is countable.
  • There exists an uncountable closed and discrete subspace of C_p(\omega_1 +1).

We discuss each of the bullet points separately.

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The function space C_p(\omega_1+1) is a dense subspace of \mathbb{R}^{\omega_1}, the product of \omega_1 many copies of \mathbb{R}. According to a result of D. P. Baturov [2], any dense normal subspace of the product of \omega_1 many separable metric spaces has countable extent (also see Theorem 1a in this previous post). Thus C_p(\omega_1+1) cannot be normal if the second bullet point above is established.

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Now we show that there exists an uncountable closed and discrete subspace of C_p(\omega_1 +1). For each \alpha with 0<\alpha<\omega_1, define h_\alpha:\omega_1 + 1 \rightarrow \left\{0,1 \right\} by:

    h_\alpha(\gamma) = \begin{cases} 1 & \mbox{if } \gamma \le \alpha \\ 0 & \mbox{if } \alpha<\gamma \le \omega_1  \end{cases}

Clearly, h_\alpha \in C_p(\omega_1 +1) for each \alpha. Let H=\left\{h_\alpha: 0<\alpha<\omega_1 \right\}. We show that H is a closed and discrete subspace of C_p(\omega_1 +1). The fact that H is closed in C_p(\omega_1 +1) is establish by the following claim.

Claim 1
Let h \in C_p(\omega_1 +1) \backslash H. There exists an open subset U of C_p(\omega_1 +1) such that h \in U and U \cap H=\varnothing.

First we get some easy cases out of the way. Suppose that there exists some \alpha<\omega_1 such that h(\alpha) \notin \left\{0,1 \right\}. Then let U=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in \mathbb{R} \backslash \left\{0,1 \right\} \right\}. Clearly h \in U and U \cap H=\varnothing.

Another easy case: If h(\alpha)=0 for all \alpha \le \omega_1, then consider the open set U where U=\left\{f \in C_p(\omega_1 +1): f(0) \in \mathbb{R} \backslash \left\{1 \right\} \right\}. Clearly h \in U and U \cap H=\varnothing.

From now on we can assume that h(\omega_1+1) \subset \left\{0,1 \right\} and that h is not identically the zero function. Suppose Claim 1 is not true. Then h \in \overline{H}. Next observe the following:

    Observation.
    If h(\beta)=1 for some \beta \le \omega_1, then h(\alpha)=1 for all \alpha \le \beta.

To see this, if h(\alpha)=0, h(\beta)=1 and \alpha<\beta, then define the open set V by V=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in (-0.1,0.1) \text{ and } f(\beta) \in (0.9,1.1) \right\}. Note that h \in V and V \cap H=\varnothing, contradicting that h \in \overline{H}. So the above observation is valid.

Now either h(\omega_1)=1 or h(\omega_1)=0. We claim that h(\omega_1)=1 is not possible. Suppose that h(\omega_1)=1. Let V=\left\{f \in C_p(\omega_1 +1): f(\omega_1) \in (0.9,1.1) \right\}. Then h \in V and V \cap H=\varnothing, contradicting that h \in \overline{H}. It must be the case that h(\omega_1)=0.

Because of the continuity of h at the point x=\omega_1, of all the \gamma<\omega_1 for which h(\gamma)=1, there is the largest one, say \beta. Now h(\beta)=1. According to the observation made above, h(\alpha)=1 for all \alpha \le \beta. This means that h=h_\beta. This is a contradiction since h \notin H. Thus Claim 1 must be true and the fact that H is closed is established.

Next we show that H is discrete in C_p(\omega_1 +1). Fix h_\alpha where 0<\alpha<\omega_1. Let W=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in (0.9,1.1) \text{ and } f(\alpha+1) \in (-0.1,0.1) \right\}. It is clear that h_\alpha \in W. Furthermore, h_\gamma \notin W for all \alpha < \gamma and h_\gamma \notin W for all \gamma <\alpha. Thus W is open such that \left\{h_\alpha \right\}=W \cap H. This completes the proof that H is discrete.

We have established that H is an uncountable closed and discrete subspace of C_p(\omega_1 +1). This implies that C_p(\omega_1 +1) is not normal.

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Remarks

The set H=\left\{h_\alpha: 0<\alpha<\omega_1 \right\} as defined above is closed and discrete in C_p(\omega_1 +1). However, the set H is not discrete in a larger subspace of the product space. The set H is also a subset of the following \Sigma-product:

    \Sigma(\omega_1)=\left\{x \in \mathbb{R}^{\omega_1}: x_\alpha \ne 0 \text{ for at most countably many } \alpha < \omega_1 \right\}

Because \Sigma(\omega_1) is the \Sigma-product of separable metric spaces, it is normal (see here). By Theorem 1a in this previous post, \Sigma(\omega_1) would have countable extent. Thus the set H cannot be closed and discrete in \Sigma(\omega_1). We can actually see this directly. Let \alpha<\omega_1 be a limit ordinal. Define t:\omega_1 + 1 \rightarrow \left\{0,1 \right\} by t(\beta)=1 for all \beta<\alpha and t(\beta)=0 for all \beta \ge \alpha. Clearly t \notin C_p(\omega_1 +1) and t \in \Sigma(\omega_1). Furthermore, t \in \overline{H} (the closure is taken in \Sigma(\omega_1)).

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Reference

  1. Arhangel’skii, A. V., Normality and Dense Subspaces, Proc. Amer. Math. Soc., 48, no. 2, 283-291, 2001.
  2. Baturov, D. P., Normality in dense subspaces of products, Topology Appl., 36, 111-116, 1990.

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\copyright \ 2014 \text{ by Dan Ma}

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