ON THE COMPLETENESS PROPERTIES OF THE C -COMPACT-OPEN TOPOLOGY ON C ( X ) 1 , 2

This is a study of the completeness properties of the space C rc ( 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, ˘ C ech-complete, pseudocomplete and almost ˘ C ech-complete.


Introduction
The set-open topology on a family λ of nonempty subsets of the Tychonoff space X (the λopen 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 ∈ C(X) : f (F ) ⊆ U }, where F ∈ λ and U is an open subset of real line R, form a subbase of the λ-open topology.
We denote the space C(X) with λ-open topology by C λ (X).Note that the set-open topology and its properties depend on the family λ.So if we take a family λ 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 λ then a topological space C λ (X) may have weaker properties, for example, it can not be a regular or Hausdorff space.
Special interest for applications when a space C λ (X) is a locally convex topological vector space (TVS).Therefore, we take a "good" family λ 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 C rc (X).
This article is a continuation of the article [3] on the study of topological properties of the space C rc (X).
The importance of studying the C-compact-open topology on C(X), due to the fact that if C λ (X) is a locally convex TVS then the family λ consists of C-compact subsets of X.
Moreover if C λ (X) is a topological (even paratopological) group then the family λ consists of C-compact subsets of X.
In [19] was found to be characteristic for the space C λ (X) such that C λ (X) is a toplogical group, TVS or locally convex TVS.

A. V. Osipov
Note that if the set-open topology coincides with the topology of uniform convergence on the family λ then C λ (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 x∈X |g(x)−f (x)| < ε}.The topology of uniform convergence on elements of a family λ (the λ-topology), where λ 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 where F ∈ λ and ε > 0, form a base of the λ-topology at a point f ∈ C(X).We denote the space C(X) with λ-topology by C λ,u (X).
Theorem 0.1.For a space X, the following statements are equivalent.
3. C λ (X) is a topological vector space.
4. C λ (X) is a locally convex topological vector space.

λ is a family of C-compact sets and λ
for any open set U of the space R}.
In [3], in addition to studying some basic properties of C rc (X), metrizability, separability and submetrizability of C rc (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 C rc (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 λ-open topology and λ-topology will be denoted as follows: [ 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 A space X is called a µ-space if every closed bounded subset of X is compact.In the literature, a µ-space is also called a hyperisocompact or a Nachbin-Shirota space (N S-space for brevity).
The closure of a set A will be denoted by A; the symbol ∅ stands for the empty set.
If A ⊆ 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 f 0 .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-Frol ík-Mrówka space in which the concepts of pseudocompactness 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]).
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 Dieudonné complete space is compact (see [14]).

Uniform Completeness of C rc (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, ε , where f ∈ C(X), F is a C-compact subset of X and ε 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 ε > 0, we define the seminorm p A on C(X) and V A,ε as follows: 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 C rc (X), we need to talk about rccontinuous functions and rc f -spaces.D e f i n i t i o n 1.1.A function f : X → R is said to be rc-continuous, if for every C-compact subset A ⊆ X, there exists a continuous function g : X → R such that g| A = f | A .A space X is called a rc f -space if every rc-continuous function on X is continuous.
Theorem 1.1.The space C rc (X) is uniformly complete if and only if X is a rc f -space.P r o o f.Note that a C-compact subset of X is a bounded set.So by Theorem 4.6 (see [14]), C rc (X) is uniformly complete.

Complete metrizability and some related completeness properties of C rc (X)
In this section, we study various kinds of completeness C rc (X).In particular, here we study the complete metrizability of C rc (X) in a wider setting, more precisely, in relation to several other completeness properties.
A space X is called Čech-complete if X is a G δ -set in βX.A space X is called locally Čechcomplete if every point x ∈ X has a Čech-complete neighborhood.Another completeness property which is implied by Čech-completeness is that of pseudocompleteness.This is space having a sequence of π-bases {B n : n ∈ N} such that whenever B n ∈ B n for each n and B n+1 ⊆ B n , then {B n : n ∈ N} = ∅ (see [20]).
In [6], it has been shown that a space having a dense Čech-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 ⊆ U .A sequence {U n } of subsets of X is said to be complete if every filter base F on X which is controlled {U n } clusters at some x ∈ X.A sequence {U n } of collections of subsets of X is called complete if {U n } is a complete sequence of subsets of X whenever U n ∈ U n 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 G δ -subset of any Hausdorff space in which it is densely embedded; (2) X has a complete sequence of open covers; (3) X is Čech-complete.
From this result, it easily follows that a Tychonoff space X is Čech-complete if and only if X is a G δ -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 Čech-complete if X has a complete sequence of open almost-covers.Every almost Čech-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 C rc (X) is a locally convex space, C rc (X) is a Baire space if and only if C rc (X) is of second category in itself.Also since a locally convex Baire space is barreled, first we find a necessary condition for C rc (X) to be barreled.A locally convex space L is called barreled if each barrel in L is a neighborhood of 0 L .Theorem 2.1.If C rc (X) is barreled, then every bounded subset of X is contained in a Ccompact subset of X. P r o o f.Let A be a bounded subset of X and let W = {f ∈ C(X) : p A (f ) ≤ 1}.Then it is routine to check that W is closed, convex, balanced and absorbing, that is, W is a barrel in C rc (X).Since C rc (X) is barreled, W is a neighborhood of f 0 and consequently there exist a closed C-compact subset P of X and ε > 0 such that f 0 , P, ε ⊆ W .We claim that A ⊆ P .If not, let x 0 ∈ A \ P .So there exists a continuous function f : X → [0, 2] such that f (x) = 0 for all x ∈ P and f (x 0 ) = 2. Clearly f ∈ f 0 , P, ε , but f / ∈ W . Hence we must have A ⊆ P .
If X is µ-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, C c (X) is barreled if X is µ-space.Hence if X is realcompact, then C rc (X) is barreled.In particular, since the Niemytzki plane L is realcompact, C rc (L) is barreled.
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 C rc (X) is not barreled.
Recall that a space X is called hemi-C-compact if there exists a sequence of C-compact subsets {A n : n ∈ N} in X such that for any C-compact subset A of X, A ⊆ A n 0 holds for some n 0 ∈ N [3].
In [3] obtained the characterization of metrizability of space C rc (X).