The aim of this note is to classify the countable (Hausdorff) compact topological spaces; namely, we prove Sierpinski-Mazurkiewicz theorem following [Stefan Mazurkiewicz and Wacław Sierpiński, Contribution à la topologie des ensembles dénombrables, Fundamenta Mathematicae 1, 17–27, 1920].
First, let us define Cantor-Bendixson rank of some topological space by transfinite induction:
- is the set of limit points of for any limit ordinal ,
- for any limit ordinal .
The Cantor-Bendixson rank of , denoted by , is the smallest ordinal satisfying .
Lemma: Let be a countable (Hausdorff) compact topological space. Then is a countable successor ordinal, say , and is finite nonempty.
Proof. For any ordinal , . Because is countable, there exists a countable ordinal such that ie. .
Let be the Cantor-Bendixson rank of . Since any perfect set in a compact space has cardinality at least (proved below), . In particular, if is a limit ordinal, is a decreasing sequence of nonempty closed sets converging to , which is impossible since is compact.
Therefore, is a successor ordinal, say , and is nonempty and finite (otherwise, would be nonempty by compactness).
Definition: Let be a countable (Hausdorff) compact topological space such that and has cardinality . We say that is the characteristic system of .
As preliminaries, we show that for all countable ordinal and integer , there exists a countable compact topological space of characteristic system , namely .
Lemma: Let be a countable ordinal and be a positive integer. Then is a countable compact topological space of characteristic system .
Proof. First, we show that is the characteristic system of by transfinite induction over . For , it is clear.
Then, writing and noticing that the closed sets are isomorphic to and that they can be separated by open sets, we deduce that .
Let be a limit ordinal. In the same way, writing and noticing that that the closed sets are isomorphic to and that they can be separated by open sets, we deduce that since .
We conclude the proof with again the same argument: writing as a disjoint union of closed sets isomorphic to and separated by open sets, we deduce that .
Now we can state and prove our main theorem:
Theorem: (Sierpinski-Mazurkiewicz) Let be a countable compact topological space of characteristic system . Then is homeomorphic to .
Proof: First, we prove that any countable compact space is metrizable:
Let be the set of continuous functions from to and let .Then is continuous and, because distinguishes points and closed sets in (since is completely regular), is injective and open, so embedds into . Taking the linear subspace spanned by , we can suppose that is a subspace of .
By compactness, is in fact a subspace of a product of closed intervals , and since each is homeomorphic to , we can view as a subspace of .
But defines a distance over , so is metrizable.
Lemma: Any metric space is isometric to a subspace of a (complete) normed space.
Proof. Let be a metric space and be the set of continuous bounded functions with sup norm. Fix . Then is an isometry from into .
Therefore, any countable compact space can be viewed as a subspace of a normed vector space.
For any countable ordinal and positive integer , let be the assertion: “Any countable compact space of Cantor-Bendixson rank such that is homeomorphic to .”
Step 1: is true.
Let be a countable compact topological space of characteristic system . Let and let be a bijection. The extension defined by is a homeomorphism.
Step 2: If is true then is true.
Let be a countable compact topological space of characteristic system and . Then, viewing as a subspace of a normed space , there exist parallel hyperplanes , …, such that has connected components , …, with .
Because is true, each is homeomorphic to . Therefore, is homeomorphic to .
Step 3: If is true for any and , then is true.
Let be a countable compact topological space of characteristic system and . Without loss of generality, we can suppose so that , ie. there exists a sequence in converging to . Then, viewing as a subspace of a normed space , there exists a sequence of postive real numbers converging to zero such that the family of spheres does not meet and has infinitely many connected components , , … with .
By assumption, each is homeomorphic to some with and . Therefore, is homeomorphic to .
First, notice that . If then because is compact; consequently, there exist and such that , hence , a contradiction with . Therefore, and is true.
Step 4: We conclude the proof by transfinite induction.
Corollary: Two countable compact topological spaces are homeomorphic if, and only if, they have the same characteristic system.
Corollary: Any countable compact topological space is homeomorphic to a well-orded set.
Corollary: Any countable compact topological space is homeomorphic to a subspace of .
Corollary: There exist countable compact topological spaces up to homeomorphism.