COUNTABLE PRODUCTS ELENA GUREVICH Abstract. In this paper, we extend our study to countably infinite products of topological spaces.. The Cantor Set Let us constract a very curios (but usefull) set known as the Cantor Set. Consider the closed unit interval [0, ] and delete from it the open interval (, 2 ) and denote the remaining closed set by G = [0, ] [2, ] Next, delete from G the open intervals ( 9, 2 9 ) and ( 7 9, 8 9 ), and denote the remaining closed set by G 2 = [0, 9 ] [2 9, ] [2, 7 9 ] [8 9, ] If we continue in this way, at each stage deleting the open middle third of each closed interval remaining from the previous stage we obtain a descending sequence of closed sets G G 2 G G n... The Cantor Set G is defined by G = n= G n, and being the intersection of closed sets, is a closed subset of [0, ]. As [0, ] is compact, the Cantor Space (G, τ),(that is, G with the subspace topology), is compact. It is useful to represent the Cantor Set in terms of real numbers written to basis, that is, ternaries. In the ternary system, 76 5 8 would be written as 22.002, since this represents 2 + 2 2 + + 0 + 0 + 0 2 + + 2 4 So a number x [0, ] is represented by the base number a a 2 a... a n... where x = n= a n n a n {0,, 2} n N Turning again to the Cantor Set G, it should be clear that an element of [0, ] is in G if and only if it can be written in ternary form with a n n N, so 2 / G 5 8 / G but G and G. [we denote 2 = 0...., = 0.02222... = 0.2222... ] Thus we have a function f from the Cantor Set into the set of all sequences of the form < a, a 2, a,..., a n, >, where each a i {0, 2} and f is one-to-one and onto. 2. The Product Topology Definition 2.. Let (X, τ ), (X 2, τ 2 ),..., (X n, τ n ),... be a countably infinite family of topological spaces. Then the product X i of the sets X i, i N consists of all the infinite sequences < x, x 2, x,..., x n, > where x i X i for all i. The product space Date: May 6, 20.
2 (X i, τ i ) consists of the product X i with the topology τ having as its basis the family B = { O i : O i τ i, and O i = X i for all but a finite number of i s} Proposition 2.2. Let (X i, τ i ), (Y i, τ i ) i N, be countably infinite families of topological spaces, having product spaces ( X i, τ) ( Y i, τ ) respectively. If the mapping h i : (X i, τ i ) (Y i, τ i ) is continuous for each i N then so is the mapping - h : ( X i, τ) ( Y i, τ ) given by h(< x, x 2, x,..., x n, >) =< h (x ), h 2 (x 2 ), h (x ),..., h n (x n ), > Proof. It suffices to show that if O is a basic open set in ( X i, τ) then h (O) is open in ( Y i, τ ). Consider the basic open set U U 2 U n Y n+... where U i τ i for i =, 2,..., n. Then h (U U 2 U n Y n+... ) = h (U ) h (U 2 ) h (U n ) X n+... since the continuity of each h i implies that h i (u i ) τ i for i =, 2,..., n. So h is continuous.. The Cantor space and the Hilbert cube Proposition.. Let (G, τ) be the Cantor space and for each i N let (A i, τ i ) be the set 0, 2 with the discrete topology, and let ( A i, τ) be it s product space. Then the map f : (G, τ) ( A i, τ ) given by f( an ) =< a n, a 2, a,..., a n, > is a homeomorphism. Proof. It is easy to see that f is one-to-one and onto. To prove that f is continuous it suffices to show for any basic given set U = U U 2 U n A n+... and any point a =< a, a 2, a,..., a n, > U there exists an open set W an such that n f(w ) U. Consider the open interval ( an, n N+2 an + ) and let W be the n N+2 intersection of this open intervals with G. Then W is open in (G, τ) and if x = x n n W then x i = a i for i =, 2,..., N. So f(x) U U 2 U n A n+... and thus f(w ) U as required. Proposition.2. Let (G i, τ i ), i N, be a countably infinite family of topological spaces each of which is homeomorphic to the Cantor space (G, τ) then (G, τ) (G i, τ i ) n (G i, τ i ) for each n N. Proof. First let us verify that (G, τ) (G, τ ) (G 2, τ 2 ). This is, by proposition. equivalent to showing that (A i, τ i ) (A i, τ i ) (A i, τ i ) where each (A i, τ i ) is the set 0, 2 with the discrete topology. Now we define a function Θ : (A i, τ i ) (A i, τ i ) (A i, τ i ) by Θ(< a, a 2, a, >, < b, b 2, b, >) =< a, b, a 2, b 2, >. It s easy to show that Θ is a homeomorphism, and so (G, τ ) (G 2, τ 2 (G, τ). By induction we get that (G, τ) (A i, τ i ) for every positive integer n. To show the infinite product case, define the mapping Φ : [ (A i, τ i ) (A i, τ i )... ] (A i, τ i ) by Φ(< a, a 2, a, >, < b, b 2, b, >, < c, c 2, c, >, < d, d 2, d, >, < e, e 2, e, > ) =< a, a 2, b, a, b 2, c, a 4, b, c 2, d, a 5, b 4, c, d 2, e, > Again it is easily verified that Φ is homeomorphism and the proof is complete.
Proposition.. The topological space [0, ] is a continuous image of the Cantor space (G, τ). Proof. It suffices to find a continuous mapping Φ of (A i, τ i ) onto [0, ]. Such a mapping is given by a i Φ(< a, a 2,..., a i, >) = 2 i+ It is easy to show that Φ is onto and continuous, and the proof is complete. Definition.4. For each positive integer n, let the topological space (I n, τ n ) be homeomorphic to [0, ]. Then the product space n= (I n, τ n ) is called Hilbert cube and is denoted by I. The product space n (I i, τ i ) is called the n-cube, and is denoted by I n, for each n N Theorem.5. The Hilbert cube is compact. Proof. By proposition. there is a continuous mapping Φ n of (G n, τ n ) onto (I n, τ n where for each n N, (G n, τ n ) and (I n, τ n are homeomorphic to the Cantor space and [0, ] respectively. Therefore, there is a continuous mapping Ψ of n= (G n, τ n ) onto n= (I n, τ n = I but (G n, τ n ) is homeomorphic to the Cantor space (G, τ). Therefore I is a continuous image of the compact space (G, τ) and hance is compact. Proposition.6. Let (X i, τ i ) i N be a countably infinite family of matrizable spaces, then (X i, τ i ) is metrizable. Corollary.7. The Hilbert cube is metrizable. 4. The Cantor space and the Hilbert cube Definition 4.. A topological space (X, τ) is said to be separable if it has a countable dense subset. Proposition 4.2. Let (X, τ) be a compact metrizable space. Then (X, τ) is separable. Proof. Let d be a metric on X which induces the topology tau. For each positive integer n, let S n be the family of all open balls having centers in X and radius n.then S n is an open covering of X and so there is a finite sub covering U n = {U n, U n2,..., U nk } for some k N. Let y nj be the center of U n, j =, 2,..., k and Y N = {y n,..., y nk }. Put Y = n= Y n, then Y is a countable subset of X. Now, if V is any non open set in (X, τ), then for any v V, V contains an open ball B of radius n. As U n is an open cover of X, v U nj for some j. Thus d(v, y nj ) < n, and so y n j B V. Hance V Y and so Y is dense in X. Corollary 4.. The Hilbert cube is separable space. Definition 4.4. A topological space (X, τ) is said to be a T space if every singelton set {x}, x X, is a closed set. Theorem 4.5. (Urisohn s Theorem) Every separable metric space (X, d) is homeomorphic to a subspace of the Hilbert cube. Proof. We need to find a countably infinite family of mappings f i : (X, d) [0, ] which are a) continuous, and b) separate points and closed sets. Without loss of generality we can assume that d(a, b) for all a and b in X, since every metric is equivalent to such a metric. As (X, d) is separable, there exists a countable dense subset Y = {y i, i N}. For
4 each i N, define f i : X [0, ] by f i (x) = d(x, y i ). It is clear that each f i continuous. To see that the mappings {f i : i N} separate points and closed sets, let x X and A be any closed set not containing x. Now X \ A is an open set, and so contains an open ball B of radius ɛ and center x, for some ɛ > 0. As Y is dense in X, there exists a y n, such that d(x, y n ) < ɛ 2, thus d(y n, a) ɛ 2 for all a A. So [0, ɛ 2 ) is an open set in [0, ] which contains f n (x), but f n (a) / [0, ɛ 2 ), for all a A. This implies f n(a) [ ɛ 2, ]. As the set [ ɛ 2, ] is closed, this implies f n (A) [ ɛ 2, ]. Hance f n(x) / f n (A) and thus the family {f i : i N} separates points and closed spaces. Corollary 4.6. Every compact metrizable space is homeomorphic to a closed subspace of the Hilbert cube. Corollary 4.7. If for each i N, (X i, τ i ) is compact metrizable space, then X i, τ i is compact and metrizable. Definition 4.8. A topological space (X, τ) is said to satisfy a the second axiom of countability (or to be second countable) if there exists a basis B for τ such that B consists of only a countable number of sets. Proposition 4.9. Let (X, d), be a matric space and τ the induced topology. Then (X, τ) is a separable space if and only if it satisfies the second axiom of countability. Proof. Let (X, τ) be separable. Then it has a countable dense subset Y = {y i : i N}. Let B consists of all the balls (in the metric d) with center y i, and radius n, for some positive integer n. Clearly B is countable, and we will show that it is a basis for τ. Let V τ. Then for any v V, V contains an open ball, B, of radius n about r. As Y is dense in X, there exists a y m Y such that d(y m, v) < 2n. Let B be the open ball with center y m and radius 2n. Then the triangle inequality implies B B V Also B B. Hance B is a basis for τ. Conversely, let (X, τ) second countable, having a countable basis B = {B i : i N}. For each B i let b i be any element of B i, and put Z equal to the set of all such b i. Then Z is countable set. Further, if V τ, then B i V for some i, and so b i V. Thus V Z Hance Z is dense in X. Consequently (X, τ) is separable. Theorem 4.0. (Urisohn s theorem and its converse) Let (X, τ) be a topological space. Then (X, τ) is separable and metrizable if and only if it is homeomorphic to a subspace of the Hilbert cube. Proof. If (X, τ) is separable and metrizable then Urisohn s theorem says that it is homeomorphic to a subspace of the Hilbert cube. Conversely, let (X, τ) be homeomorphic to the subspace (Y, τ ) of the Hilbert cube I, I is separable and metrizable, hance it is second countable and metrizable (since any subspace of a second countable and metrizable space is second countable and metrizable) (Y, τ ) is second countable and metrizable. Therefor it is separable. Hance (X, τ) is also separable and metrizable. 5. Peano s Theorem Proposition 5.. Every separable metrizable space (X, τ ) is a continuous image of a subspace of the cantor space (G, τ). Further if (X, τ ) is compact, then the subspace can be chosen to be closed in (G, τ). Proposition 5.2. Let (Y, τ ) be a (non empty) closed subspace of the Cantor space (G, τ). Then there exists a continuous mapping of (G, τ) onto (Y, τ ). Theorem 5.. Every compact metrizable space is a continuous image of the Cantor space.
Proposition 5.4. let f be a continuous mapping of a compact metric space (X, d) onto a Hausdorff space (Y, τ ). Then (Y, τ ) is compact and metrizable. Theorem 5.5. (Peano) For each positive integer n, there exists a continuous mapping φ n of [0, ] onto the n-cube I n. Proof. There exists a continuous mapping Φ n of the Cantor space (G, τ) onto the n-cube I n. As (G, τ) is obtained from [0, ] by successively dropping out middle thirds, we extend Φ n to a continuous mapping Ψ n : [0, ] I n by defining Ψ n to be linear on each omitted interval, that is, if (a.b) is one of the open intervals comprising [0, ] G, then Ψ n is defined on (a.b) by Ψ n (αa + ( α)b) = αφ n (a) + ( αφ n (b)) 0 α It is easily verified that Ψ n is continuous. 5. S. Morris, Topology without tears, 2007. References