This is a discussion on a compact space called Helly space. The discussion here builds on the facts presented in Counterexample in Topology . Helly space is Example 107 in . The space is named after Eduard Helly.
Let be the closed unit interval with the usual topology. Let be the set of all functions . The set is endowed with the product space topology. The usual product space notation is or where each . As a product of compact spaces, is compact.
Any function is said to be increasing if for all (such a function is usually referred to as non-decreasing). Helly space is the subspace consisting of all increasing functions. This space is Example 107 in Counterexample in Topology . The following facts are discussed in .
- The space is compact.
- The space is first countable (having a countable base at each point).
- The space is separable.
- The space 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 stems from the fact that is a closed subspace of the compact space .
Additional facts of concerning Helly space are discussed.
- The product space is normal.
- Helly space contains a copy of the Sorgenfrey line.
- Helly space is not hereditarily normal.
The space is the space of all countable ordinals with the order topology. Recall is the product space . The product space is Example 106 in . This product is not normal. The non-normality of is based on this theorem: for any compact space , the product is normal if and only if the compact space is countably tight. The compact product space is not countably tight (discussed here). Thus is not normal. However, the product is normal since Helly space is first countable.
To see that contains a copy of the Sorgenfrey line, consider the functions defined as follows:
for all . Let . Consider the mapping defined by . With the domain having the Sorgenfrey topology and with the range being a subspace of Helly space, it can be shown that is a homeomorphism.
With the Sorgenfrey line embedded in , the square contains a copy of the Sorgenfrey plane , 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, is the product space where and is Helly space consisting of all increasing functions in . Consider the following two subspaces of .
The subspace is a closed subset of , hence compact. We claim that subspace is separable and has a closed and discrete subset of cardinality continuum. This means that the subspace is not a normal space.
First, we define a discrete subspace. For each with , define as follows:
Let . The set as a subspace of is discrete. Of course it is not discrete in since is compact. In fact, for any , (closure taken in ). However, it can be shown that is closed and discrete as a subset of .
We now construct a countable dense subset of . To this end, let be a countable base for the usual topology on the unit interval . For example, we can let be the set of all open intervals with rational endpoints. Furthermore, let be a countable dense subset of the open interval (in the usual topology). For convenience, we enumerate the elements of and .
We also need the following collections.
For each and for each with , we would like to arrange the elements in increasing order, notated as follow:
For the set , we have . For the set , is to the left of for . Note that elements of are pairwise disjoint. Furthermore, write . If , then . If , then .
For each and as detailed above, we define a function as follows:
The following diagram illustrates the definition of when both and have 4 elements.
Let be the set of over all and . The set is a countable set. It can be shown that is dense in the subspace . In fact is dense in the entire Helly space .
To summarize, the subspace is separable and has a closed and discrete subset of cardinality continuum. This means that is not normal. Hence Helly space 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).
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 , the cube is not hereditarily normal (see Theorem 3 in this post). Thus a non-normal subspace is found in , or . In fact, for any compact non-metric space , an excellent exercise is to find where a non-normal subspace can be found. Is it in , the square of or the cube of ? In the case of Helly space , a non-normal subspace can be found in .
A natural question is: is there a compact non-metric space such that both and are hereditarily normal and 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 is likely to fail hereditarily normality in either or . See here for a discussion of this set-theoretic question.
- Kelly, J. L., General Topology, Springer-Verlag, New York, 1955.
- Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
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