GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 565-570

ON CONTINUOUS EXTENSIONS

LAWRENCE NARICI AND EDWARD BECKENSTEIN

Abstract. We consider various possibilities concerning the continuous extension of continuous functions taking values in an ultrametric space. In Section 1 we consider Tietze-type extension theorems concerning continuous extendibility of continuous functions from compact and closed subsets to the whole space. In Sections 2 and 3 we consider extending “separated” continuous functions in such a way that certain continuous extensions remain separated. Functions taking values in a complete ultravalued field are dealt with in Section 2, and the real and complex cases in Section 3.

1. A Tietze-Type Extension Theorem

An ultranormal topological space T is a Hausdorff space in which disjoint closed subsets may be separated by clopen sets. As shown by R. Ellis ([1], cf. [2], Th. 8.27, p. 258) the following version of the Tietze extension theorem obtains in the nonarchimedean setting:

(Bounded) continuous functions x mapping a closed subset K of the ultranormal space T into a complete separable metric space have (bounded) continuous extensions to all of T.

We prove a similar result below—we eliminate the separability requirement on the domain T but consider an ultrametric space Y as the codomain. Thus, the result below is not more general, just different. For the sake of the proof we define a clopen n-partition of a subset M of T with respect to x : M → Y to be a finite pairwise disjoint collection of clopen subsets V_{i} of T such that M ⊂ ⋃_{i=1}^{n} V_{i} and for all s, t ∈ V_{i} ∩M, d (x (s) , x (t)) ≤ 1/n. As in many of the classical constructions of functions with certain properties (continuous but nowhere differentiable, for example) we create a sequence of functions each of which almost has the desired property, whose uniform limit does.

565

1991 Mathematics Subject Classification. 46S10, 54C45, 54D35.

Key words and phrases. Continuous extension, Tietze extension theorem.

1072-947X/96/1100-0565$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Theorem 1.1. Let y be a point of the complete ultrametric space (Y, d). In each part below y could belong to K.

(a) A continuous map x of a compact subset K of the ultraregular space T into Y may be continuously extended to x̄ defined on all of T with x̄ (T) = x (K) ∪ {y}.

(b) A continuous map x of a closed subset K of the ultranormal space T into Y with relatively compact range may be continuously extended to x̄ defined on all of T with x̄ (T) ⊂ [cl x (K)]∪{y} where “cl” denotes topological closure.

Proof. (a) Let C (p, r) denote the closed ball of radius r > 0 about p ∈ Y. For each t ∈ K, let B(t, 1/n) = {s ∈ K : d (x(s), x(t)) ≤ 1/n} = x^{−1} (C (x (t) , 1/n)). Since K is compact, there exist t_{1}, …, t_{n_{1}} ∈ K such that K = ⋃_{i=1}^{n_{1}} B (t_{i}, 1). In the standard way, we may rewrite the relatively clopen subsets of K, B (t_{i}, 1), 1 ≤ i ≤ n_{1}, to get a pairwise disjoint relatively clopen cover {A_{i}} of K with the property that, for any 1 ≤ i ≤ n_{1} and s, t ∈ A_{i}, d (x (s) , x (t)) < 1. We continue to assume that the A_{i} are nonempty and that t_{i} ∈ A_{i} for 1 ≤ i ≤ n_{1}; otherwise, we would reindex the sets and choose some new points t_{i} as needed. Since T is ultraregular and K is compact, we may choose pairwise disjoint clopen subsets U_{i} of T such that U_{i} ∩ K = A_{i} for each i = 1, 2, …, n_{1}. Evidently the family {U_{i}} is a clopen 1-partition of K with respect to x. We define the continuous “step” function

x_{1} (t) = \{ \begin{matrix}x (t_{i}) , & t ∈ U_{i}, 1 ≤ i ≤ n_{1}, \\ y, & t ∉ ⋃_{i=1}^{n_{1}} U_{i}. \end{matrix}

For any t ∈ K, for some 1 ≤ i ≤ n_{1}, t ∈ U_{i} and x_{1} (t) = x (t_{i}) ; hence

d (x_{1} (t) , x (t)) ≤ max [d (x_{1} (t) , x (t_{i})) , d (x (t_{i}) , x (t))]
= max [d (x (t_{i}) , x (t_{i})) , d (x (t_{i}) , x (t))]
= d (x (t_{i}) , x (t))
≤ 1.

It follows that d_{K} (x_{1}, x) = sup{d(x_{1} (t) , x (t)) : t ∈ K} ≤ 1. For n = 2, we decompose each A_{i} into a clopen 2-partition with respect to x. On A_{1}, for example, we get nonempty disjoint clopen subsets B_{1}, …, B_{r_{1}} of T such that each is a subset of U_{1} and (⋃_{i=1}^{r_{1}} B_{i}) ∩ K = A_{1}. Choose s_{i} ∈ B_{i} ∩ A_{1} for 1 ≤ i ≤ r_{1} and define

x_{2} (t) = \{ \begin{matrix}x (s_{i}) , & t ∈ B_{i}, 1 ≤ i ≤ r_{1}, \\ x_{1} (t) = x(t_{1}), & t ∉ U_{1} − ⋃_{i=1}^{r_{1}} B_{i}, \\ y, & t ∉ ⋃_{i=1}^{n_{1}} U_{i} \end{matrix}

and define x_{2} in a similar way on the rest of the A_{i}. In so doing, we create a continuous step function x_{2} : T → Y such that d_{K} (x_{2}, x) ≤ 1/2 and d_{T} (x_{1}, x_{2}) ≤ 1. We continue in this fashion to construct a sequence (x_{n}) of functions continuous on T such that, for every n, d_{K} (x_{n}, x) ≤ 1/n and d_{T} (x_{n}, x_{n−1}) ≤ 1/ (n − 1). We denote the pointwise limit of (x_{n}) as x̄. As x_{n} → x̄ uniformly, it follows that x̄ is continuous; clearly x̄ = x on K.

(b) As cl x (K) is compact, there exist y_{1}, y_{2}, …, y_{n_{1}} ∈ cl x (K) such that the C (y_{i}, 1) cover cl x (K). Consequently K = ⋃_{i=1}^{n_{1}} x^{−1} [C (y_{i}, 1)]. Since T is ultranormal there are pairwise disjoint clopen sets U_{i} such that x^{−1} [C (y_{i}, 1)] ⊂ U (i, 1) for 1 ≤ i ≤ n_{1}. Choose t_{i} ∈ x^{−1} [C (y_{i}, 1)] for 1 ≤ i ≤ n_{1} and define x_{1} as was done in (a) and continue the process to obtain the sequence (x_{n}) and the limit x̄.

Corollary 1.2. Let T, y and Y be as in the theorem. (a) Let x and w be continuous functions defined on the disjoint compact subsets K and L of T taking values in Y. For any disjoint clopen supersets U and V of K and L, respectively, x and w can be continuously extended to x̄ and w̄, respectively, defined on T, with x̄ (T) ⊂ x (K) ∪ {y} , w̄ (T) ⊂ w (L) ∪ {y} and x (t) = w (t) = y on C(U ∪ V).

(b) If x and w are continuous functions with relatively compact range defined on the disjoint closed subsets K and L of the ultranormal space T taking values in Y, then x and w can be continuously extended to x̄ and w̄, respectively, defined on T, with x̄ (T) ⊂ cl x (K)∪{y} , w̄ (T) ⊂ cl w (L)∪{y} and x (t) = w (t) = y on C(U ∪ V).

Proof. (a) By the theorem, x and w may be continuously extended to U and V, respectively. It only remains to define those extensions to assume the value y on C(U ∪ V) to obtain the desired extensions to all of T. The result of (b) follows directly from part (b) of the theorem.

For any topological space T and any semigroup G we say that two functions x, y : T → G are strongly separated if cl coz x ∩ cl coz y = ∅, and separated if coz x ∩ coz y = ∅, where coz x denotes the cozero set of x. Note that x and y are separated if and only xy = 0. As has been developed in a number of papers ([3]–[8]) there is ample reason for interest in additive maps H : C (T) → C (S) between spaces of scalar-valued continuous functions C (T) and C (S) which are [weakly] separating in the sense that [strongly] separated functions x and y have separated images Hx and Hy. For that reason it is of importance to know when continuous functions which are separated or weakly separated on a subset have continuous extensions which remain separated, the subject of the corollary below; the point y of the theorem is taken to be the zero element of the semigroup.

Corollary 1.3. Let T be ultraregular and let Y be a semigroup on which a complete ultrametric is defined (not necessarily a topological semigroup).

(a) If let x and w are strongly separated continuous functions defined on a compact subset K of T then there are separated continuous extensions x̄ and w̄ defined on T such that x̄w̄ = 0.

(b) Two strongly separated continuous functions x and w with relatively compact range defined on a closed subset K of the ultranormal space T have continuous extensions x̄ and w̄ defined on T such that x̄w̄ = 0.

2. The Case of Complete Valued Fields

Let T be ultraregular and let C(T, F) denote the space of continuous functions taking values in the complete, nonarchimedean nontrivially valued field F. We show in Theorem 2.2 that pairwise strongly separated functions x_{1}, x_{2}, …, x_{n} ∈ C(K, F) have pairwise separated extensions x̄_{1}, x̄_{2}, …, x̄_{n} ∈ C(T, F) for two types of subset K.

Lemma 2.1. Let K be a closed subset of the compact ultraregular space T and let x_{1}, x_{2}, …, x_{n} be elements of C(T, F) which are pairwise separated on K. Then there exist pairwise separated elements x̄_{1}, x̄_{2}, …, x̄_{n} in C(T, F) such that x̄_{i} | K = x_{i} | K for i = 1, …, n. (If the x_{i} were only defined on K then they could be continuously extended to T by Corollary 1.3.)

Proof. Assume that x, y ∈ C(T, F) are separated on K. Let M bound x and y. Let G(x, i) = x^{−1} {a ∈ F : |a| ≤ M/i} for each positive integer i, with G(y, i) defined analogously. Let U(x, i) = G(x, i) − G(x, i + 1) and let U(y, i) be defined similarly. Let C(x, i) and C(y, i) denote the intersections of U(x, i) and U(y, i), respectively, with K. Note that the unions of C(x, i) and C(y, i), i ∈ N, yield coz x and coz y, respectively. Since x and y are separated on K, C(x, i) and C(y, j) are disjoint for all positive integers i and j. Let G(i) denote the clopen set G(x, i) ∩ G(y, i) for each i. Since x and y are separated on K, C(x, m) and C(y, m) are disjoint subsets of G(i) for all m ≥ i; C(x, m) and C(y, m) are subsets of CG(i), the complement of G(i), for all m < i. Thus there exist pairwise disjoint clopen sets V (x, i) and V (y, i) such that C(x, i) ⊂ V (x, i) ⊂ U(x, i) and C(y, i) ⊂ V (y, i) ⊂ U(y, i). Let k(x, i) and k(y, i) denote the characteristic functions of V (x, i) and V (y, i), respectively. The functions x_{m} = ∑_{i=1}^{m} k(x, i)x and y_{m} = ∑_{i=1}^{m} k(y, i)y are uniform Cauchy sequences on T. Call their limits x̄ and ȳ in C(T, F), respectively. Since x_{m} and y_{m} are separated for each m, x̄ and ȳ are separated as well; the restrictions of x̄ and ȳ to K are just x and y, as is easy to verify. The method for extending this to a finite number of functions—rather than just two—is a simple modification of this idea.

In the next result we drop the assumption of compactness and get an extension theorem.

Theorem 2.2. For compact K and ultraregular T [or closed K and ultranormal T] and pairwise strongly separated functions x_{1}, x_{2}, …, x_{n} ∈ C(K, F) [with relatively compact range], there exist pairwise separated extensions x̄_{1}, x̄_{2}, …, x̄_{n} ∈ C(T, F) of x_{1}, x_{2}, …, x_{n}.

Proof. It suffices to consider two separated functions x and y in C(K, F). Let β_{0}T denote the Banaschewski compactification of T, and continuously extend x and y to βx and βy in C(β_{0}T, F) by Corollary 1.3; then replace βx and βy by functions x̄ and ȳ as described in the preceding lemma. Finally, restrict x̄ and ȳ to T.

3. The Real and Complex Case

Let K be a compact subset of the Tihonov space T, and let C(K, F) and C(T, F) denote the spaces of F -valued functions on K and T, where F = R or C. Let H be a finite subset of C(K, F) which is pairwise separated in the sense that xy = 0 for each x and y in H. We show in Theorem 3.3 that each x in H has an extension x̄ in C(T, F) such that the set H̄ of extensions is pairwise separated. We begin with two lemmas about nonnegative functions.

Lemma 3.1. If the functions in H are nonnegative then each x in H has an extension x̄ ∈ C(T, R) such that the set H̄ of such extensions is pairwise separated.

Proof. We proceed by induction. We begin by considering the case in which H has two elements x and y. Let βv denote the continuous extension of the difference v = x − y to the Stone-Cech compactification βT of T. Let βv^{+} and βv^{−} denote the positive and negative parts of βv, respectively, so that βv = βv^{+} − βv^{−}. Since x and y are nonnegative and xy = 0, βv^{+} and βv^{−} are continuous extensions of x and y, respectively; clearly (βv^{+})(βv^{−}) = 0. It only remains to restrict βv^{+} and βv^{−} to T to complete the argument in this case. Now suppose that H has n + 1 ≥ 3 elements and choose x_{1}, x_{2}, …, x_{n} from H. By the induction hypothesis, there exist continuous, separated extensions w_{1}, w_{2}, …, w_{n} of x_{1}, x_{2}, …, x_{n} to T. We now repeatedly pair x_{n+1} with each of the w_{i}, i = 1, …, n, to get separated continuous extensions (by the first part of the argument) z_{i} of x_{i} and v_{i} of x_{n+1} for i = 1, …, n. Now let x̄_{i} = inf(w_{i}, z_{i}) for i = 1, …, n and let x̄_{n+1} = inf {v_{1}, …, v_{n}}. These are the desired extensions of x_{1}, x_{2}, …, x_{n}.

In the next result we assume that each x is separated from each y but do not suppose that the x’s are a pairwise separated set, nor are the y’s.

Lemma 3.2. Let x_{1}, x_{2}, …, x_{n} and y_{1}, y_{2}, …, y_{n} denote families of nonnegative functions in C(K) such that x_{i} is separated from y_{j} for all i and j. Then there exist continuous extensions x̄_{i} and ȳ_{i}, i = 1, …, n, of these functions to T with the same property. Proof. By Lemma 3.1 there exist separated extensions x(i, j) and y(i, j) of x_{i} and y_{j}, respectively, for each i each j. The functions x̄_{i} = inf{x(i, j) : 1 ≤ j ≤ n} and ȳ_{j} = inf {y(i, j) : 1 ≤ i ≤ n} are separated on all of T for all i and j.

Theorem 3.3. With F = ℝ or ℂ, and H as before Lemma 3.1, then each x in H has an extension x̄ in C(T, F) such that the set H̄ of extensions is pairwise separated. Proof. Let F = R and let x and y be separated members of C(K, ℝ). The positive and negative parts x^{+}, x^{−}, y^{+} and y^{−} of x and y, respectively, are pairwise separated. Since they are nonnegative, they can be extended to a pairwise separated collection \overline{x^{+}}, \overline{x^{−}}, \overline{y^{+}} and \overline{y^{−}} on T as in Lemma 3.1. The desired extensions are now given by x̄ = \overline{x^{+}} − \overline{x^{−}} and \overline{y} = \overline{y^{+}} − \overline{y^{−}}. It is straightforward to extend this to the case of more than two functions. To handle the case where the field is ℂ, let x and y be separated continuous functions on K. Each of the functions (Re x)^{+}, (Re x)^{−}, (Im x)^{+}, (Im x)^{−} is separated from each of the functions (Re y)^{+}, (Re y)^{−}, (Im y)^{+}, (Im y)^{−}on K and Lemma 3.2 can be applied to these functions. The separated extensions to T of those functions can be used to construct separated extensions of x and y.

References

1. R. L. Ellis, Extending continuous functions on zero-dimensional spaces. Math. Annal. 186(1970), 114–122.

2. J. B. Prolla, Topics in functional analysis over valued division rings. North-Holland Math. Studies 77, North- Holland, New York, 1982.

3. J. Araujo, E. Beckenstein, and L. J. Narici, Biseparating maps and homeomorphic real compactifications. J. Math. Anal. Appl. 192(1995), 258–265.

4. E. Beckenstein and L. Narici, A non-Archimedean Stone–Banach theorem. Proc. Amer. Math. Soc. 100(1987), 242–246.

5. E. Beckenstein and L. Narici, Automatic continuity of certain linear isomorphisms. Acad. Roy. Belg. Bull. Cl. Sci. 73(1987), 191–200.

6. E. Beckenstein, L. Narici, and A. Todd, Automatic continuity of linear maps on spaces of continuous functions. Manuscripta Math. 62(1988), 257– 275.

7. S. Hernandez, E. Beckenstein, and L. Narici, Banach–Stone theorems and separating maps. Manuscripta Math. 86(1995), 409–416.

8. L. Narici, E. Beckenstein, and J. Araujo, Separating maps on rings of continuous functions. p-Adic functional analysis, ed. by N. de Grande-de Kimpe, S. Navarro, and W. Schikhof, 69–82, Ed. Univ. Santiago de Chile, Santiago, 1994.

(Received 08.07.1994) Authors’ addresses:

Lawrence Narici

St. John’s University

Jamaica, NY 11439 USA Edward Beckenstein St. John’s University Staten Island, NY 10301 USA