In this paper we continue to study the property of separability of function space C(X) with the open-point and bi-point-open topologies. We show that for every perfect Polish space X a set C(X) with the bipoint-open topology is separable. We also show in the iterated perfect set model that for every regular space X with countable network a set C(X) with bi-point-open topology is separable iff a dispersion character (X) = c.
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In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the International Youth School-conference ¾SoProMat-2017¿, Yekaterinburg, Russia, 06-Feb-2017, published at http://ceur-ws.org
In [
A set A I = [a; b] R.
If Ch(X) is a separable space, then X has a -network consisting of I-sets. (Theorem 2.3.)
X be called I-set if there is a continuous function f 2 C(X) such that f (A) contains an interval In this paper we use the following conventions. The symbols R, P, Q and N denote the space of real numbers, irrational numbers, rational numbers and natural numbers, respectively. Recall that a dispersion character (X) of X is the minimum of cardinalities of its nonempty open subsets.
Recall also that a space be called Polish space if it is a separable complete metrizable space.
By a set of reals we mean a zero-dimensional, separable metrizable space every non-empty open set which has the cardinality the continuum. 2
Note that if the space Ch(X) is a separable space then (X) c. If A = ffig is a countable dense set of Ch(X) then for each non-empty open set U of X we have S fi(U ) = R. It follows that jU j c.
Also note that if the space Cph(X) is a separable space then Cp(X) is a separable space and Ch(X) is separable. It follows that X is a separable submetrizable (coarser separable metric topology) space and (X) = c.
Recall that a family of subsets of a space X is called T0-separating if whenever x and y are distinct points of X, there exists V 2 containing exactly one of the points x and y.
In [
Theorem 2.1. If X is a Tychonoff space with network consisting non-trivial connected sets, then the following are equivalent.
In the present paper, we consider more general form this theorem.
Theorem 2.2. If X is a Tychonoff space with -network consisting non-trivial connected sets, then the following are equivalent.
Доказательство. (1) ) (2). Let Cph(X) be a separable space. There is a countable dense subset A = ffig of the space Cph(X). Fix = fBj g some a countable base for R consisting of bounded open intervals.
Consider = ff 1(B) : f 2 A; B 2 g.
We show that the family is required family.
Let x1 and x2 be distinct points of X. Consider an open base set Q = [fx1g; (c1; d1)]+ T[fx2g; (c2; d2)]+ of the space Cph(X) where (ci; di) 2 for i = 1; 2 and (c1; d1) T (c2; d2) = ;. There is h 2 Q T A. Clearly that h 1((ci; di)) 2 , xi 2 h 1((ci; di)) for i = 1; 2 and h 1((c1; d1)) T h 1((c2; d2)) = ;.
The countable family is required family.
(2) ) (3). Let = fZig be the countable family with required conditions. We can assume that is closed under finite unions. For each i 2 N there is a continuous function fi : X 7! I = [0; 1] such that Zi = fi 1(0).
Let Ii = I fig for every i 2 N. By letting (x; i1)E(y; i2) whenever x = 0 = y or x = y and i1 = i2 we define an equivalence relation E on the set Si2N Ii.
The formula ([(x; i1)]; [(y; i2)]) = jx yj; if i1 = i2;
x + y; if i1 6= i2;
defines a metric on the set of equivalence classes of E. This space - as well as the corresponding metrizable
space - be called the metrizable hedgehog of spininess @0 and be denoted J (!) (Example 4.1.5 in [
Note that for every i 2 N the mapping ji of the interval I to J (!) defined by letting ji(x) = [(x; i)] is a homeomorphic embedding. The family of all balls with rational radii around points of the form [(r; i)], where r is a rational number, is a base for J (!); so that J (!) is a separable metrizable space.
The formula hi(x) = ji(fi(x)) defines a continuous mapping hi : X 7! J (!). Note that the family fhigi1=1 is functionally separates points of X. Really let x and y be distinct points of X. There exists Z 2 containing exactly one of the points x and y and there are i0 2 N and continuous function fi0 : X 7! I = [0; 1] such that Z = fi0 1(0). Hence hi0 (x) 6= hi0 (y). Thus diagonal mapping h = 4i2Nhi : X 7! J (!)! is a continuous one-to-one mapping from X into the separable metrizable space J (!)!.
It follows that X is a separable submetrizable space.
(3) ) (1). Let X be a separable submetrizable space, i.e. X has coarser separable metric topology 1 and be -network of X consisting non-trivial connected sets. Let = fBig be a countable base of (X; 1). We can assume that closed under finite union of its elements.
For each finite family fBsi gid=1 such that Bsi T Bsj = ; for i 6= j and i; j 2 1; d and fpigid=1 Q we fix f = fs1;:::;sd;p1:::;pd 2 C(X) such that f (Bsi ) = pi for each i = 1; d.
Let G be the set of functions fs1;:::;sd;p1:::;pd where si 2 N and pi 2 Q for i 2 N. We claim that the countable set G is a dense set of Cph(X).
By proposition 2.2 in [
Choose Bsl 2 for l = 1; m + n such that 1. Bsl1 T Bsl2 = ; for l1 6= l2 and l1; l2 2 1; n + m; 2. xi 2 Bsl for l 2 1; m; 3. Bsl T Uk 6= ; for l 2 m + 1; n + m and k = l m.
Choose Bs0l 2 for l 2 1; m such that xi 2 Bs0l and Bs0l Bsl .
Choose Ak 2 for k 2 1; n such that Ak (Uk T Bsl ) where l = k + m.
Choose different points sk; tk 2 Ak for every k = 1; m.
Let S; T 2 such that S T T = ;, Bl T S = ;, Bl T T = ; for l 2 1; m and sk 2 S and tk 2 T for all k = 1; m. Fix points vi 2 (Vi T Q) for i 2 1; m.
Choose p; q 2 Q such that p < minfri : i = 1; ng and q > maxfri : i = 1; ng.
Let
8 p f or f (x) = < q f or : vl f or x 2 S x 2 T x 2 Bs0 l where l 2 1; m.
Note that f 2 W T G. This proves theorem.
Theorem 2.3. (Theorem 2.4 in [
2. X has a countable -network consisting of I-sets.
The next result is the corollary of Theorem 2.3, but we notes its as theorem due to the importance in the class of separable metrizable spaces.
2. X has a countable -network consisting of I-sets.
We have already noted that if a space Cph(X) is a separable space then
X has a -network consisting of I-sets.
Theorem 2.6. If X is a separable submetrizable space with a countable -network consisting of I-sets, then
Доказательство. The proof analogously to the proof of the implication ((2) ) (1)) in Theorem 2.4 ([
Let S = fSig be a countable -network of X consisting of I-sets. By definition of I-sets, for each Si 2 S there is the continuous function hi 2 C(X) such that hi(Si) contains an interval [ai; bi] of real line. Consider a countable set fhi;p;q(x) = api qbi hi(x) + p api qbi aig of continuous functions on X, where i 2 N, p; q 2 Q. Note that if hi(x) = ai then hi;p;q(x) = p and if hi(x) = bi then hi;p;q(x) = q.
Let = fBj g be a countable base of (X; 1) where 1 is a separable metraizable topology on X because of X is a separable submetrizable space. For each pair (Bj ; Bk) such that Bj Bk define continuous functions and for each v 2 Q hi;p;q;j;k(x) = dj;k;v(x) = hi;p;q(x) f or x 2 Bj 0 f or x 2 X n Bk: v 0 f or f or x 2 Bj x 2 X n Bk:
Let G be the set of finite sum of functions hi;p;q;j;k and dj;k;v where i; j; k 2 N and p; q; v 2 Q. We claim that the countable set G is a dense set of Cph(X).
By proposition 2.2 in [
W = [x1; V1]+ T ::: T[xm; Vm]+ T[U1; r1] T ::: T[Un; rn] be a base set of Cph(X) where n; m 2 N, xi 2 X, Vi is an open set of R for i 2 1; m, Uj is an open set of X and rj 2 R for j 2 1; n and for i 6= j, xi 6= xj and Ui T Uj = ;.
Fix points yj 2 Uj for j = 1; n and choose Bsl 2 for l = 1; n + m such that Bsl1 T Bsl2 = ; for l1 6= l2 and l1; l2 2 1; n + m and xi 2 Bsl for l 2 1; m and yj 2 Bsl for l 2 m + 1; n. Choose Bs0l 2 for l 2 1; m such that xi 2 Bs0l and Bs0l Bsl and choose Bs0l 2 for l 2 m + 1; n + m such that yj 2 Bs0l Bsl where l = j + m.
Fix points vi 2 (Vi T Q) for i 2 1; m and pj ; qj 2 Q such that pj < rj < qj for j = 1; n.
Consider g 2 G such that g = ds01;s1;v1 + ::: + ds0m;sm;vm + hi1;p1;q1;s0m+1;sm+1 + ::: + hin;pn;qn;s0m+n;sn+m where Sik and l = k + m.
Note that g 2 W T G. This proves theorem.
Bs0 T Uk for k = 1; n
l
Corollary 2.7. If X is a perfect Polish space, then Cph(X) is separable.
Доказательство. It follows immediately from fact that any regular closed subset of a space X is a perfect
Polish space and it contains some set which is homeomorphic to 2! ([
X hasn’t countable chain condition, hence, X hasn’t countable -network consisting of I-sets; X is a separable submetrizable space;
Cph(X) is a separable space.
Example 2.8. (Example 4.3. in [
be a direct sum of real lines R.
Question 1. Assume that X is a separable submetrizable space with uncountable -network consisting of I-sets. Does Cph(X) is separable ?
Recall that a set of reals X is null if for each positive there exists a cover fIngn2N of X such that Pn diam(In) < . A set of reals X has strong measure zero if, for each sequence f ngn2N of positive reals, there exists a cover fIngn2N of X such that diam(In) < n for all n. For example, every Lusin set has strong measure zero.
In [
In [
In [
In the present paper, we consider more general form this theorem.
Theorem 2.11. ( the iterated perfect set model) If X is a regular space with a countable network, then the following are equivalent.
3. X has a countable -network consisting of I-sets. Доказательство. (1) ) (2). Note that if the space Cph(X) is a separable space then Cp(X) is a separable space and Ch(X) is separable. It follows that X is a separable submetrizable space and, hence, (X) = c. (2) ) (3). Let (X) = c.
(I). We show that any separable metrizable space M of cardinality c is I-set of M , i.e. there exists a continuous function f : M 7! R such that f (M ) I.
Really, if a real-valued continuous image of space M has cardinality less c for any f 2 C(M ), then M is a zero-dimensional space. It follows that M is a set of reals and, by the iterated perfect set model, there is a continuous map from this set onto the closed unit interval I.
If there is a real-valued continuous image of space M such that it has cardinality c, then either it contains an interval I or it is a set of reals and, again, by the iterated perfect set model, there is a continuous map from this set onto the closed unit interval I.
(II). Recall that a regular space with a countable network is normal and a separable submetrizable space. Since (X) = c and X is a regular space with a countable network, it follows that X has countable -network consisting of closed sets of cardinality c. We show that is required -network.
Let f be a condensation from X onto a separable metrizable space. Fix A 2 and consider the mapping h = f A. By point (I), h(A) is I-set of h(A), i.e. there exist the continuous function f : h(A) 7! R such that f (h(A)) I. Since X is a normal space, by Tietze-Urysohn Extension Theorem, the mapping f h can be extended to a real-valued continuous map F : X 7! R. Note that F (A) = f (h(A)) I i.e. A is I-set of X.
(2) ) (3). It follows from Theorem 2.6.
Remark 2.12. The main results of this paper were announced in:
https://arxiv.org/abs/1604.04609. Since then, several remarkable articles ([