Helly Space

This is a discussion on a compact space called Helly space. The discussion here builds on the facts presented in Counterexample in Topology [2]. Helly space is Example 107 in [2]. The space is named after Eduard Helly.

Let I=[0,1] be the closed unit interval with the usual topology. Let C be the set of all functions f:I \rightarrow I. The set C is endowed with the product space topology. The usual product space notation is I^I or \prod_{t \in I} W_t where each W_t=I. As a product of compact spaces, C=I^I is compact.

Any function f:I \rightarrow I is said to be increasing if f(x) \le f(y) for all x<y (such a function is usually referred to as non-decreasing). Helly space is the subspace X consisting of all increasing functions. This space is Example 107 in Counterexample in Topology [2]. The following facts are discussed in [2].

  • The space X is compact.
  • The space X is first countable (having a countable base at each point).
  • The space X is separable.
  • The space X has an uncountable discrete subspace.

From the last two facts, Helly space is a compact non-metrizable space. Any separable metric space would have countable spread (all discrete subspaces must be countable).

The compactness of X stems from the fact that X is a closed subspace of the compact space C.

Further Discussion

Additional facts of concerning Helly space are discussed.

  1. The product space \omega_1 \times X is normal.
  2. Helly space X contains a copy of the Sorgenfrey line.
  3. Helly space X is not hereditarily normal.

The space \omega_1 is the space of all countable ordinals with the order topology. Recall C is the product space I^I. The product space \omega_1 \times C is Example 106 in [2]. This product is not normal. The non-normality of \omega_1 \times C is based on this theorem: for any compact space Y, the product \omega_1 \times Y is normal if and only if the compact space Y is countably tight. The compact product space C is not countably tight (discussed here). Thus \omega_1 \times C is not normal. However, the product \omega_1 \times X is normal since Helly space X is first countable.

To see that X contains a copy of the Sorgenfrey line, consider the functions h_t:I \rightarrow I defined as follows:

    \displaystyle  h_t(x) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ 0 \le x \le t \\            \text{ } & \text{ } \\          \displaystyle  1 &\ \ \ \ \ \ t<x \le 1 \\                                 \end{array} \right.

for all 0<t<1. Let S=\{ h_t: 0<t<1 \}. Consider the mapping \gamma: (0,1) \rightarrow S defined by \gamma(t)=h_t. With the domain (0,1) having the Sorgenfrey topology and with the range S being a subspace of Helly space, it can be shown that \gamma is a homeomorphism.

With the Sorgenfrey line S embedded in X, the square X \times X contains a copy of the Sorgenfrey plane S \times S, which is non-normal (discussed here). Thus the square of Helly space is not hereditarily normal. A more interesting fact is that Helly space is not hereditarily normal. This is discussed in the next section.

Finding a Non-Normal Subspace of Helly Space

As before, C is the product space I^I where I=[0,1] and X is Helly space consisting of all increasing functions in C. Consider the following two subspaces of X.

    Y_{0,1}=\{ f \in X: f(I) \subset \{0, 1 \} \}

    Y=X - Y_{0,1}

The subspace Y_{0,1} is a closed subset of X, hence compact. We claim that subspace Y is separable and has a closed and discrete subset of cardinality continuum. This means that the subspace Y is not a normal space.

First, we define a discrete subspace. For each x with 0<x<1, define f_x: I \rightarrow I as follows:

    \displaystyle  f_x(y) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ 0 \le y < x \\           \text{ } & \text{ } \\          \displaystyle  \frac{1}{2} &\ \ \ \ \ y=x \\            \text{ } & \text{ } \\          \displaystyle  1 &\ \ \ \ \ \ x<y \le 1 \\                                 \end{array} \right.

Let H=\{ f_x: 0<x<1 \}. The set H as a subspace of X is discrete. Of course it is not discrete in X since X is compact. In fact, for any f \in Y_{0,1}, f \in \overline{H} (closure taken in X). However, it can be shown that H is closed and discrete as a subset of Y.

We now construct a countable dense subset of Y. To this end, let \mathcal{B} be a countable base for the usual topology on the unit interval I=[0,1]. For example, we can let \mathcal{B} be the set of all open intervals with rational endpoints. Furthermore, let A be a countable dense subset of the open interval (0,1) (in the usual topology). For convenience, we enumerate the elements of A and \mathcal{B}.

    A=\{ a_1,a_2,a_3,\cdots \}

    \mathcal{B}=\{B_1,B_2,B_3,\cdots \}

We also need the following collections.

    \mathcal{G}=\{G \subset \mathcal{B}: G \text{ is finite and is pairwise disjoint} \}

    \mathcal{A}=\{F \subset A: F \text{ is finite} \}

For each G \in \mathcal{G} and for each F \in \mathcal{A} with \lvert G \lvert=\lvert F \lvert=n, we would like to arrange the elements in increasing order, notated as follow:

    F=\{t_1,t_2,\cdots,t_n \}

    G=\{E_1,E_2,\cdots,E_n \}

For the set F, we have 0<t_1<t_2< \cdots <t_n<1. For the set G, E_i is to the left of E_j for i<j. Note that elements of G are pairwise disjoint. Furthermore, write E_i=(p_i,q_i). If 0 \in E_1, then E_1=[p_1,q_1)=[0,q_1). If 1 \in E_n, then E_n=(p_n,q_n]=(p_n,1].

For each F and G as detailed above, we define a function L(F,G):I \rightarrow I as follows:

    \displaystyle  L(F,G)(x) = \left\{ \begin{array}{ll}                     \displaystyle  t_1 &\ \ \ \ \ 0 \le x < q_1 \\           \text{ } & \text{ } \\          \displaystyle  t_2 &\ \ \ \ \ q_1 \le x < q_2 \\           \text{ } & \text{ } \\          \displaystyle  \vdots &\ \ \ \ \ \vdots \\           \text{ } & \text{ } \\          \displaystyle  t_{n-1} &\ \ \ \ \ q_{n-2} \le x < q_{n-1} \\           \text{ } & \text{ } \\          \displaystyle  t_n &\ \ \ \ \ q_{n-1} \le x \le 1 \\                                             \end{array} \right.

The following diagram illustrates the definition of L(F,G) when both F and G have 4 elements.

Figure 1 – Member of a countable dense set

Let D be the set of L(F,G) over all F \in \mathcal{A} and G \in \mathcal{G}. The set D is a countable set. It can be shown that D is dense in the subspace Y. In fact D is dense in the entire Helly space X.

To summarize, the subspace Y is separable and has a closed and discrete subset of cardinality continuum. This means that Y is not normal. Hence Helly space X is not hereditarily normal. According to Jones’ lemma, in any normal separable space, the cardinality of any closed and discrete subspace must be less than continuum (discussed here).

Remarks

The preceding discussion shows that both Helly space and the square of Helly space are not hereditarily normal. This is actually not surprising. According to a theorem of Katetov, for any compact non-metrizable space V, the cube V^3 is not hereditarily normal (see Theorem 3 in this post). Thus a non-normal subspace is found in V, V \times V or V \times V \times V. In fact, for any compact non-metric space V, an excellent exercise is to find where a non-normal subspace can be found. Is it in V, the square of V or the cube of V? In the case of Helly space X, a non-normal subspace can be found in X.

A natural question is: is there a compact non-metric space V such that both V and V \times V are hereditarily normal and V \times V \times V is not hereditarily normal? In other words, is there an example where the hereditarily normality fails at dimension 3? If we do not assume extra set-theoretic axioms beyond ZFC, any compact non-metric space V is likely to fail hereditarily normality in either V or V \times V. See here for a discussion of this set-theoretic question.

Reference

  1. Kelly, J. L., General Topology, Springer-Verlag, New York, 1955.
  2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

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