Jones’ Lemma

Jones’ lemma is a great tool in working with normal spaces. It is useful, under cerntain conditions, to show the non-normality of a space. The lemma establishes an upper bound for the cardinality of closed and discrete sets in any separable and normal space. Thus, whenever you have a separable space with a closed and discrete set whose cardinality exceeds the upper bound, you have a non-normal space (see examples discussed below). One way to prove the Jones’ lemma is to explore the set-theoretic relationship between the density (the minimum cardinality of a dense set) and the cardinality of closed and discrete sets in normal spaces. We sketch a proof of this lemma and give some examples. We also state an extension of Jones’ lemma. All spaces under consideration are at least Hausdorff.

Let X be a space. A subset D of X is said to be a closed and discrete set in X if D is a closed set in X and D, in the relative topology, is a discrete space. Good basic references are [1] and [3]. For more detailed information about cardinal functions, see [2].
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Jones’ Lemma
Let X be a separable and normal space. Then for any set D that is a closed and discrete set in X, we have \displaystyle 2^{\lvert D \lvert} \le 2^{\omega}.

Corollary 1
Let X be a separable and normal space. Then for any set D that is a closed and discrete set in X, we have \displaystyle \lvert D \lvert < 2^{\omega}.

Jones' lemma, as stated above, is essentially saying that the cardinality of continuum is an upper bound of the cardinalities of the power sets of closed and discrete sets in any separable and normal space. The corollary says that the cardinality of continuum is an upper bound of the cardinalities of closed and discrete sets in any separable and normal space. As indicated at the beginning, the corollary is a great way for checking the non-normality of a separable space.

We now give a sketch of the proof Jones' lemma. Let C(X) be the set of all continuous real-valued functions defined on the space X. Suppose X is a separable space. A key point is that the cardinality of C(X) is sandwiched between the two cardinalities in the lemma:

\displaystyle (1) \ \ \ \ \ 2^{\lvert D \lvert} \le \lvert C(X) \lvert \le 2^{\omega}

The first inequality in (1) says that there are at least as many continuous real-valued functions as there are subsets of the closed and discrete set D. To see this, we appeal for some help from Urysohn's lemma. For each E \subset D, E and D-E are disjoint closed sets in X. By Urysohn's lemma, there is a continuous function f_E:X \rightarrow [0,1] such that f_E maps E to 1 and f_E maps D-E to 0. Note that the mapping \Psi: \mathcal{P}(D) \rightarrow C(X) defined by \Psi(E)=f_E is a one-to-one map, where \mathcal{P}(D) is the collection of all subsets of D.

The second inequality in (1) says that there are at most continuum many continuous real-valued functions defined on the space X. In other words, the number of continuous functions is capped by the cardinality continuum (actually equals continuum in this case, but one inequality is all we need here). Let G be a countable dense subset of X. Consider the map \rho:C(X) \rightarrow \mathbb{R}^{G} defined by \rho(f)=f \upharpoonright G, which is the function f restricted to the set G. The notation \mathbb{R}^{G} refers to the set of all functions from the set G into \mathbb{R}.

The key point here is that any continuous functions f:X \rightarrow \mathbb{R} and g:X \rightarrow \mathbb{R}, if they agree on the countable dense set G (f \upharpoonright G=g \upharpoonright G), then f=g on the whole space X.So \rho is a one-to-one map. Some elementary cardinal arithmetic shows that \lvert \mathbb{R}^G \lvert=2^\omega. Thus the second inequality in (1) is established.

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Examples

Let S be the Sorgenfrey line. This is the real line with the topology generated by half open intervals of the form [a,b). The Sorgenfrey line is a classic example of a normal space whose square is not normal. The Sorgenfrey plane S \times S is separable with a closed and discrete set of cardinality continuum. By the corollary of Jones’ lemma, the Sorgenfrey plane is not normal. The Sorgenfrey line is discussed in greater details in this post.

The tangent disc space is another example of a separable space with a closed and discrete set of size continuum, hence ensuring that it is a non-normal space.

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Generalization

The generalization involves the notion of density and the notion of extent. The density of an infinite Hausdorff space X is the smallest cardinal number of the form \lvert G \lvert where G is a dense subset of X. This cardinal number is denoted by d(X). For any separable space X, we have d(X)=\omega. If the space X is finite, then the convention adopted by many authors is that d(X)=\omega.

The extent of a space X is the smallest infinite cardinal \mathcal{K} such that every closed and discrete set in X has cardinality \le \mathcal{K}. The extent of the space X is denoted by e(X). For more detailed information about cardinal functions, see [2].

The following are the generalization of Jones’ lemma.

Jones’ Lemma
Let X be a normal space. Then for any set D that is a closed and discrete set in X, we have \displaystyle 2^{\lvert D \lvert} \le 2^{d(X)}.

Corollary 2
Let X be a normal space. Then for any set D that is a closed and discrete set in X, we have \displaystyle \lvert D \lvert < 2^{d(X)}, which implies \displaystyle e(X) \le 2^{d(X)}.

Corollary 2 suggests that the cardinal number 2^{d(X)} is an upper bound of the cardinalities of closed and discrete sets in any normal space X. As a result of this, the cardinal number 2^{d(X)} dominates the extent e(X) in a normal space.
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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Hodel, R., Cardinal Functions I, Handbook of Set-Theoretic Topology, (K. Kunen and J. E. Vaughan, eds), 1984, Elsevier Science Publishers B. V., Amsterdam, 1-61.
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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