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10URAL MATHEMATICAL JOURNAL, Vol. 1, No. 1, 2015
ON THE COMPLETENESS PROPERTIES OF THE C-COMPACT-OPEN TOPOLOGY ON C(X)1,2
Alexander V. Osipov
N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and Ural Federal University, Ekaterinburg, Russia,
OAB@list.ru
This is a study of the completeness properties of the space Crc(X) of continuous real-valued functions on a Tychonoff space X, where the function space has the C-compact-open topology. Investigate the properties such as completely metrizable, Cech-complete, pseudocomplete and almost Cech-complete.
Keywords: C-compact-open topology, Set-open topology, Cech-complete, Baire space, Function space.
Introduction
The set-open topology on a family A of nonempty subsets of the Tychonoff space X (the A-open topology) is a generalization of the compact-open topology and of the topology of pointwise convergence. This topology was first introduced by Arens and Dugundji [7].
All sets of the form {f e C(X) : f (F) C U}, where F e A and U is an open subset of real line R, form a subbase of the A-open topology.
We denote the space C(X) with A-open topology by C\(X). Note that the set-open topology and its properties depend on the family A. So if we take a family A of all finite, compact or pseudocompact subsets of X then we get point-open, compact-open, pseudocompact-open topology on C(X) respectively. These topologies actively studied and find their application in measure theory and functional analysis.
Sure if we take an arbitrary family A then a topological space Ca(X) may have weaker properties, for example, it can not be a regular or Hausdorff space.
Special interest for applications when a space Ca(X) is a locally convex topological vector space (TVS). Therefore, we take a "good" family A of subsets of X which define locally convex TVS on C(X). For example a families of all compact, finite, metrizable compact, sequentially compact, countable compact, pseudocompact or C-compact subsets of X are "good" families (see [18]).
Recall that a subset A of a space X is called C-compact subset X if, for any real-valued function f continuous on X, the set f (A) is compact in R.
Note that, in the case A = X, the property of the set A to be C-compact coincides with the pseudocompactness of the space X.
The space C(X), equipped with the set-open topology on the family of all C-compact subsets of X, is denoted by Crc(X).
This article is a continuation of the article [3] on the study of topological properties of the space
Crc(X ).
The importance of studying the C-compact-open topology on C(X), due to the fact that if C\(X) is a locally convex TVS then the family A consists of C-compact subsets of X.
Moreover if Ca(X) is a topological (even paratopological) group then the family A consists of C-compact subsets of X.
In [19] was found to be characteristic for the space C\(X) such that C\(X) is a toplogical group, TVS or locally convex TVS.
*The work was supported by Project 15-16-1-6 comprehensive program of UB RAS. 2Published in Russian in Trudy Inst. Mat. i Mekh. UrO RAN, 2012. Vol. 18. No 2. P. 191-198.
Note that if the set-open topology coincides with the topology of uniform convergence on the family A then C\(X) is a topological algebra.
Recall that the topology of uniform convergence is given by a base at each point f £ C(X). This
base consists of all sets {g £ C(X) : sup \g(x)-f (x)\ < e). The topology of uniform convergence on
xex
elements of a family A (the A-topology), where A is a fixed family of non-empty subsets of the set X, is a natural generalization of this topology. All sets of the form {g £ C(X) : sup \g(x) — f (x)\ < e),
xeF
where F £ A and e > 0, form a base of the A-topology at a point f £ C(X). We denote the space C(X) with A-topology by Cx,u(X).
In [19] proved the following theorem (Theorem 3.3).
Theorem 0.1. For a space X, the following statements are equivalent.
1. C\(X) = C\,U(X).
2. C\(X) is a topological group.
3. C\(X) is a topological vector space.
4. C\(X) is a locally convex topological vector space.
5. A is a family of C-compact sets and A = A(C), where A(C) = {A £ A : for every C-compact subset B of the space X with B a A, the set [B,U] is open in C\(X) for any open set U of the space R).
In [3], in addition to studying some basic properties of Crc(X), metrizability, separability and submetrizability of Crc(X) have been studied. In this paper, we study various kinds of completeness of the C-compact topology such as complete metrizability, Cech-completeness, pseudocompleteness and almost Cech-completeness of Crc(X).
Throughout the rest of the paper, we use the following conventions. All spaces are completely regular Hausdorff, that is, Tychonoff.
The elements of the standard subbases of the A-open topology and A-topology will be denoted as follows:
[F, U] = {f £ C(X) : f(F) C U),
(f, F, e) = {g £ C(X) : sup \f (x) — g(x)\ < e), where F £ A, U is an open subset of R and
xeF
e > 0.
If X and Y are any two spaces with the same underlying set, then we use X = Y, X < Y and X < Y to indicate, respectively, that X and Y have same topology, that the topology Y is finer than or equal to the topology on X and that the topology on Y is strictly finer than the topology on X. The symbols R and N denote the spaces of real numbers and natural numbers, respectively.
We recall that a subset of X that is the complete preimage of zero for a certain function from C(X) is called a zero-set. A subset O of a space X is called functionally open (or a cozero-set) if X \ O is a zero-set.
Cover is called functionally open if it consists of functionally open subsets of X. Let G C C(X). A set A C X is said to be G-bounded if f (A) is a bounded subset of R for each f £ G. We say that A is bounded in X if A is G-bounded for G = C(X).
A space X is called a ^-space if every closed bounded subset of X is compact. In the literature, a ^-space is also called a hyperisocompact or a Nachbin-Shirota space (AS*-space for brevity). The closure of a set A will be denoted by A; the symbol 0 stands for the empty set. If A C X and f £ C(X), then we denote by f\a the restriction of the function f to the set A. As usual, f (A) and f _1(A) are the image and the complete preimage of the set A under the mapping f, respectively.
The constant zero function defined on X is denoted by f0. We call it the constant zero function in C(X).
The remaining notation can be found in [5].
Obvious that a pseudocompact subset of X is a C-compact subset of X and a C-compact subset of X is a bounded subset of X by definition.
In [5] given a well-known Isbell-FroKfc-Mrowka space in which the concepts of pseudocompact-ness and C-compactness differ even for closed subsets.
We note some important properties of C-compact subset (see [2] and [8]).
The subset A is an C-compact subset of X if and only if every countable functionally open (in X) cover of A has a proper subcollection whose union is dense in A.
For any Tychonoff space X, pseudocompactness equivalent to feebly compactness of X. Recall that a space X is called a feebly compact if whenever countably infinite locally finite open cover of X has a proper subcollection whose union is dense in X. It is well known that the closure of the pseudocompact (bounded) subset of X will be pseudocompact (bounded) subset of X. It holds true for C-compact set [1].
Note that for a closed subset A in a normal Hausdorff space X, the following equivalent (see [14]).
1. A is countably compact.
2. A is pseudocompact.
3. A is C-compact subset of X.
4. A is bounded.
Recall that a Tychonoff space X is called submetrizable if X admits a weaker metrizable topology.
Note that for a subset A in a submetrizable space X, the following are equivalent (see [4]).
1. A is countably compact subset of X.
2. A is pseudocompact subset of X.
3. A is sequentially compact subset of X.
4. A is C-compact subset of X.
5. A is compact subset of X.
6. A is metrizable compact subset of X.
Note that every closed bounded subset of Dieudonne complete space is compact (see [14]).
1. Uniform Completeness of Crc(X)
There are three ways to consider the C-compact-open topology on C (X) [3].
First, one can use as subbase the family {[A, V] : A is a C-compact subset of X and V is an open subset of R}. But one can also consider this topology as the topology of uniform convergence on the C-compact subsets of X, in which case the basic open sets will be of the form (f, F, e), where f e C(X), F is a C-compact subset of X and e is a positive real number.
The third way is to look at the C-compact-open topology as a locally convex topology on C(X). For each C-compact subset A of X and e > 0, we define the seminorm pa on C(X) and Va>£ as follows: pA(f) = sup{|f (ж)|: ж e A} and Va,£ = {f e C(X): pA(f) < e}. Let Ф = {Va,£ : A is a C-compact subset of X, e > 0}. Then for each f e C(X), f + Ф = {f + V: V e Ф} forms a neighborhood base at f. This topology is locally convex since it is generated by a collection of seminorms and it is same as the C-compact-open topology on C (X). It is also easy to see that this topology is Tychonoff.
The topology of uniform convergence on the C-compact subsets of X is actually generated by the uniformity of uniform convergence on these subsets. Recall that a uniform space E is called complete provided that every Cauchy net in E converges to some element in E.
In order to characterize the uniform completeness of Crc(X), we need to talk about rc-continuous functions and rcf-spaces.
Definition 1.1. A function f : X ^ R is said to be rc-continuous, if for every C-compact subset A C X, there exists a continuous function g : X ^ R such that g\A = f\a. A space X is called a rcf-space if every rc-continuous function on X is continuous.
Theorem 1.1. The space Crc(X) is uniformly complete if and only if X is a rcf -space.
Proof. Note that a C-compact subset of X is a bounded set. So by Theorem 4.6 (see [14]), Crc(X) is uniformly complete. □
2. Complete metrizability and some related completeness properties of Crc(X)
In this section, we study various kinds of completeness Crc(X). In particular, here we study the complete metrizability of Crc(X) in a wider setting, more precisely, in relation to several other completeness properties.
A space X is called Cech-complete if X is a Gs-set in f3X. A space X is called locally Cech-complete if every point x £ X has a Cech-complete neighborhood. Another completeness property which is implied by Cech-completeness is that of pseudocompleteness.
This is space having a sequence of ^-bases {Bn : n £ N) such that whenever Bn £ Bn for each n and Bn+i C Bn, then f|{Bn : n £ N) = 0 (see [20]).
In [6], it has been shown that a space having a dense Cech-complete subspace is pseudocomplete and a pseudocomplete space is a Baire space.
Let F and U be two collections of subsets of X. Then F is said to be controlled by U, if for each U £U, there exists F £ F such that F C U .A sequence {Un) of subsets of X is said to be complete if every filter base F on X which is controlled {Un) clusters at some x £ X .A sequence {Un) of collections of subsets of X is called complete if {Un) is a complete sequence of subsets of X whenever Un £ Un for all n £ N. It has been shown in [11, Theorem 2.8] that the following statements are equivalent for a Tychonoff space X:
(1) X is a Gs-subset of any Hausdorff space in which it is densely embedded;
(2) X has a complete sequence of open covers;
(3) X is Cech-complete.
From this result, it easily follows that a Tychonoff space X is Cech-complete if and only if X is a Gs-subset of any Tychonoff space in which it is densely embedded.
We call a U of subsets of X an almost-cover of X if U is dense in X. We call a space almost Cech-complete if X has a complete sequence of open almost-covers. Every almost Cech-complete space is a Baire space, see [16, Proposition 4.5].
The property of being a Baire space is the weakest one among the completeness properties we consider here. Since Crc(X) is a locally convex space, Crc(X) is a Baire space if and only if Crc(X) is of second category in itself. Also since a locally convex Baire space is barreled, first we find a necessary condition for Crc(X) to be barreled. A locally convex space L is called barreled if each barrel in L is a neighborhood of 0^.
Theorem 2.1. If Crc(X) is barreled, then every bounded subset of X is contained in a C-compact subset of X.
Proof. Let A be a bounded subset of X and let W = {f £ C(X) : pA(f) < 1). Then it is routine to check that W is closed, convex, balanced and absorbing, that is, W is a barrel in Crc(X). Since Crc(X) is barreled, W is a neighborhood of f0 and consequently there exist a closed C-compact subset P of X and e > 0 such that (f0, P, e) C W. We claim that A C P. If not, let xo £ A \ P. So there exists a continuous function f : X ^ [0,2] such that f (x) = 0 for all x £ P and f (x0) = 2. Clearly f £ (f0, P, e), but f £W. Hence we must have A C P. □
If X is ^-space, then every closed bounded (C-compact) subset of X is a compact and consequently the C-compact-open and compact-open topologies on C(X) coincide. But by famous Nachbin-Shirota theorem, Cc(X) is barreled if X is ^-space. Hence if X is realcompact, then Crc(X) is barreled. In particular, since the Niemytzki plane L is realcompact, Crc(L) is barreled.
But there are space X such that Crc(X) is not barreled.
Example 1. (Dieudonne Plank) Let X = [0,wi] x [0,w0] \ {(w^w0)}.
Topology т is generated from the base: all the points of set [0,w1) x [0, u0) are isolated, and sets of the form Ua(@) = {(в, 7) : a < 7 ^ w0} and Va(@) = {(7, в) : a <7 ^ }.
Let A = {(^1,n) : 0 ^ n <ш0}. Take an arbitrary C-compact subset B of the space X. Since a set {a} x [0, w0] is a clopen (and hence functionally open) for any a < w1, then set ([0, w1] x {в}) П B consists of more than finite number of points for any в ^
It follows that B is a compact subset of X. In [14] was proved that the set A is a closed bounded subset of X. Since A is not compact subset of X and each C-compact subsets of X is a compact then Crc(X) is not barreled.
Recall that a space X is called hemi-C-compact if there exists a sequence of C-compact subsets {An : n e N} in X such that for any C-compact subset A of X, A С An0 holds for some n0 e N [3].
In [3] obtained the characterization of metrizability of space Crc(X).
Theorem 2.2. For any space X, the following are equivalent.
1. Crc(X) is metrizable.
2. Crc (X) is of first countable.
3. Crc(X) is of countable type.
4. Crc(X) is of pointwise countable type.
5. Crc(X) has a dense subspace of pointwise countable type.
6. Crc(X) is an M-space.
7. Crc(X) is a q-space.
8. X is hemi-C-compact.
The following theorem gives a characterization of complete metrizable of the space Crc(X).
Theorem 2.3. For any space X, the following assertions are equivalent.
1. Crc(X) is a completely metrizable.
2. Crc(X) is Cech-complete.
3. Crc(X) is locally Cech-complete.
4. Crc(X) is an open continuous image of a paracompact Cech-complete space.
5. Crc(X) is an open continuous image of a Cech-complete space.
6. X is a hemi-C-compact rcf -space.
Proof. We have earlier noted that Crc(X) is completely metrizable if and only if it is uniform complete and metrizable. Hence by Theorem 1.1 and by Theorem 2.2 , (1) ^ (6). Note that (1) ^ (2) ^ (3) and (1) ^ (4) ^ (5). Also (3) ^ (5), see [5, 3.12.19.(d)].
(5) ^ (1) A Cech-complete space is of pointwise countable type and the property of being pointwise countable type is preserved by open continuous maps. Hence Crc(X) is of pointwise countable type and consequently by Corollary 5.2 [3], Crc(X) is metrizable and hence Crc(X) is paracompact. So by Pasynkov's theorem [5, Theorem 5.5.8 (b)], Crc(X) is Cech-complete. But a Cech-complete metrizable space is completely metrizable. □
Note that proof of Theorem 2.3 is similar to that of Theorem 3.3 in [15] on a complete metrizable of space C(X) with the pseudocompact-open topology.
For studying the properties of pseudocomplete and almost Cech-complete we need to embed the space Crc(X) in a larger locally convex function space.
Let RC(X) = {f £ Rx : f\a is continuous for each C-compact subset A of X). As in case of Crc(X), we can define C-compact-open topology on RC(X). In particular, this is is a locally convex Hausdorff topology on RC (X) generated by the family of seminorms {pa : A is a C-compact subset of X), where for f £ RC(X), pA(f) = sup{\f (x)\ : x £ A).
We denote the space RC(X) with the C-compact-open topology by RCrc(X). It is clear that Crc(X) is a subspace of RCrc(X). Moreover the proof of the following result it immediate.
Theorem 2.4. If every closed C-compact subset of X is C-embedded in X, then C(X) is dense in RCrc(X).
In next theorem, the term a-space refers to a space having a a-locally finite network. Every metrizable space is a a-space.
Theorem 2.5. For a space X, consider the following conditions.
1. Crc(X) is completely metrizable.
2. Crc(X) is a pseudocomplete a-space.
3. Crc(X) is a pseudocomplete q-space.
4. Crc(X) contains a dense completely metrizable subspace.
5. Crc(X) contains a dense Cech-complete subspace.
6. Crc(X) is almost Cech-complete.
Then (1) ^ (2) ^ (3) ^ (4) ^ (5) ^ (6)
Proof. (1) ^ (2) and (4) ^ (5). These are immediate.
(2) ^ (3). A Baire space, which is a a-space as well, has a dense metrizable subspace, see [43]. So if Crc(X) is a pseudocomplete a-space, then it contains a dense metrizable space. Since every metrizable space is of pointwize countable type, by Theorem 2.2, Crc(X) is a q-space.
(3) ^ (4) If Crc(X) is a q-space, then by Theorem 2.2, Crc(X) is metrizable. But a metrizable space is pseudocomplete if and only if it contains a dense completely metrizable subspace, see [6, Corollary 2.4.].
(5) ^ (6) follows from [16, Propositions 4.4, 4.7]. □
Remark. If Crc(X) is only assumed to be pseudocomplete, it may not be almost Cech-complete. Let S be an uncountable space in which all points are isolated except for a distinguished point s, a neighborhood of s being any set containing s whose complement is countable [12, 4N].
It can be easily shown that S is a normal space. Since every C-compact subset of S is finite, the C-compact-open topology on C(S) coincides with the point-open topology on C(S). Since S is uncountable, Cp(S) is not metrizable. But Cp(S) is pseudocomplete, since every countable subset in a P-space is closed. But since Cp(X) is not metrizable, by Theorem 5.7 [13], it is not almost Cech-complete either.
Theorem 2.6. If every closed C-compact subset X is C-embedded in X, then the following assertions are equivalent.
1. Crc(X) is completely metrizable.
2. Crc(X) is almost Cech-complete.
3. X is a hemi-C-compact rcf -space.
Proof. We only need to show that (2) ^ (3). If Crc(X) is almost Cech-complete, then Crc(X) contains a dense Cech-complete subspace G. Since every closed C-compact subset of X is C-embedded in X, C(X) is dense in RCrc(X) and consequently G is dense in RCrc(X). Now since RCrc(X) contains a dense Baire subspace G, RCrc(X) is itself a Baire space. Also since G is Cech-complete, G is a Gs-set in RCrc(X).
Note that every rc-continuous function on X is in RCrc(X). In order to show that X is a rcf -space, we will show that RC (X) = C (X). So let f £ RC (X). Define the map Tf : RCrc(X) ^
RCrc(X) by Tf (g) = f + g for all g e RC(X). Since RCrc(X) is a locally convex space, Tf is a homeomorphism and consequently Tf (G) is a dense -subset of RCrc(X). Since RCrc(X) is a Baire space, Gf|Tf(G) = 0. Let h e Gf)Tf(G). Then there exists g e G such that h = f + g. So f = g - h e C (X). □
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