# Ascoli-type theorems in the cone metric space setting

## Abstract

We give some necessary and sufficient conditions for (global) continuity of the limit of a pointwise convergent net of cone metric space-valued functions, defined on a Hausdorff topological space, in terms of weak filter exhaustiveness. In this framework, we prove some Ascoli-type theorems, considering also possibly asymmetric and extended real-valued distance functions. Furthermore, we pose some open problems.

MSC:26E50, 28A12, 28A33, 28B10, 28B15, 40A35, 46G10, 54A20, 54A40, 06F15, 06F20, 06F30, 22A10, 28A05, 40G15, 46G12, 54H11, 54H12, 47H10.

## 2 Preliminaries

We begin with some fundamental properties of convergence and continuity in the lattice group context.

A nonempty set $Λ=(Λ,≥)$ is said to be directed iff ≥ is a reflexive and transitive binary relation on Λ, such that for any two elements $λ 1 , λ 2 ∈Λ$ there is $λ 0 ∈Λ$ with $λ 0 ≥ λ 1$ and $λ 0 ≥ λ 2$.

A cone metric space is a nonempty set R endowed with a function $ρ:R×R→Y$, where Y is a Dedekind complete lattice group, satisfying the following axioms:

• $ρ( r 1 , r 2 )≥0$ and $ρ( r 1 , r 2 )=0$ if and only if $r 1 = r 2$;

• $ρ( r 1 , r 2 )=ρ( r 2 , r 1 )$ (symmetric property);

• $ρ( r 1 , r 3 )≤ρ( r 1 , r 2 )+ρ( r 2 , r 3 )$ (triangular property), for all $r j ∈R$, $j=1,2,3$

(see also [11, 12]). Such a function ρ will be called a distance function. If ρ satisfies only the first and the third of the above axioms, but not necessarily the symmetric property, then we say that ρ is an asymmetric distance function and that $(R,ρ)$ is an asymmetric cone metric space (for literature on the asymmetric case, see also [34] and the related bibliography). Note that any Dedekind complete $(ℓ)$-group Y is a cone metric space: indeed, it is enough to take $ρ( y 1 , y 2 )=| y 1 − y 2 |$, $y 1 , y 2 ∈Y$ (the absolute value).

When R is a semigroup and $Y=R$, we say that R is a metric semigroup. An example of metric semigroup which is not a group is the set $L(R)$ of the fuzzy numbers (see also [1, 4]).

Example 2.1 Let T any arbitrary nonempty set, , and fix a positive real number $a≠1$. For every $t∈T$ and $f 1 , f 2 ∈R$, put

and set $d( f 1 , f 2 )= sup t ∈ T d t ( f 1 (t), f 2 (t))$. It is not difficult to check that d is an asymmetric distance function (see also [34]).

Let R be a (possibly asymmetric) cone metric space and Y be its associated Dedekind complete $(ℓ)$-group. A sequence $( σ p ) p$ in Y is called an $(O)$-sequence iff it is decreasing and $⋀ p σ p =0$. A net $( x λ ) λ ∈ Λ$ in R (that is an indexed system of elements of R such that the index set Λ is directed) is forward order convergent or forward $(O)$-convergent (resp. backward order convergent or backward $(O)$-convergent) to $x∈R$ iff there exists an $(O)$-sequence $( σ p ) p$ in Y such that for every $p∈N$ there is $λ∈Λ$ with $ρ(x, x ζ )≤ σ p$ (resp. $ρ( x ζ ,x)≤ σ p$) for all $ζ∈Λ$, $ζ≥λ$. We say that $( x λ ) λ ∈ Λ$ order converges or $(O)$-converges to $x∈R$ iff it is both forward and backward $(O)$-convergent to x, and in this case we will write $(O) lim λ ∈ Λ x λ =x$.

Let X be a Hausdorff topological space. A function $f:X→R$ is said to be forward (resp. backward) continuous at a point $x∈X$ iff there exists an $(O)$-sequence $( σ p ) p$ in Y (depending on x) such that for every $p∈N$ there is a neighborhood $U x$ of x with $ρ(f(x),f(z))≤ σ p$ (resp. $ρ(f(z),f(x))≤ σ p$) whenever $z∈ U x$.

A function $f:X→R$ is globally forward (resp. backward) continuous on X iff there is an $(O)$-sequence $( σ p ) p$ in Y such that for any $p∈N$ and $x∈X$ there is a neighborhood $U x$ of x with $ρ(f(x),f(z))≤ σ p$ (resp. $ρ(f(z),f(x))≤ σ p$) for each $z∈ U x$. We say that $f∈ R X$ is (globally) continuous on X iff it is both (globally) forward and (globally) backward continuous on X.

We now recall some basic notions on ideals and filters.

Let Λ be any nonempty set, and $P(Λ)$ be the class of all subsets of Λ. A family of sets $I⊂P(Λ)$ is called an ideal of Λ iff $A∪B∈I$ whenever $A,B∈I$ and for each $A∈I$ and $B⊂A$ we get $B∈I$. A class of sets $F⊂P(Λ)$ is a filter of Λ iff $A∩B∈F$ for all $A,B∈F$ and for every $A∈F$ and $B⊃A$ we have $B∈F$.

An ideal (resp. a filter ) of Λ is said to be non-trivial iff $I≠∅$ and $Λ∉I$ (resp. $F≠∅$ and $∅∉F$).

Let $(Λ,≥)$ be a directed set. A non-trivial ideal of Λ is said to be $(Λ)$-admissible iff $Λ∖ M λ ∈I$ for each $λ∈Λ$, where $M λ :={ζ∈Λ:ζ≥λ}$.

A non-trivial filter of Λ is $(Λ)$-free iff $M λ ∈F$ for every $λ∈Λ$.

Given an ideal of Λ, we call dual filter of the family $F={Λ∖I:I∈I}$. In this case we say that is the dual ideal of and we get $I={Λ∖F:F∈F}$.

When $Λ=N$ endowed with the usual order, the $(N)$-admissible ideals and the $(N)$-free filters are called simply admissible ideals and free filters, respectively. The filter $F cofin$ is the filter of all subsets of whose complement is finite, and its dual ideal $I fin$ is the family of all finite subsets of . The filter $F st$ is the filter of all subsets of having asymptotic density 1, while its dual ideal $I st$ is the family of all subsets of , having null asymptotic density. Note that $F st$ is a P-filter, namely a filter of such that for every sequence $( A n ) n$ in there is another sequence $( B n ) n$ in , such that the symmetric difference $A n Δ B n$ is finite for all $n∈N$ and $⋂ n = 1 ∞ B n ∈F$ (see also [33, 47]).

A nonempty family $B ′ ⊂P(Λ)$ is said to be a filter base of Λ iff for every $A,B∈ B ′$ there is an element $C∈ B ′$ with $C⊂A∩B$. Note that, if $B ′$ is a filter base of Λ, then the family is a filter of Λ. We call it the filter generated by $B ′$.

If $B ′ ={ M λ :λ∈Λ}$, then $B ′$ is a filter base of Λ, and the filter $F Λ$ generated by $B ′$ is a $(Λ)$-free filter of Λ (see also [48]).

Let be a $(Λ)$-free filter of Λ and choose $y ¯ ∈R$. A net $( s λ ) λ ∈ Λ$ in R is said to be -forward bounded (resp. -backward bounded) with respect to $y ¯$ iff there is $k 0 ∈Y$, $k 0 ≥0$ such that ${λ∈Λ:ρ( y ¯ , s λ )≤ k 0 }∈F$ (resp. ${λ∈Λ:ρ( s λ , y ¯ )≤ k 0 }∈F$). We say that $( s λ ) λ$ is -bounded with respect to $y ¯$ iff it is both -forward and -backward bounded with respect to $y ¯$, and that $( s λ ) λ$ is bounded (resp. forward bounded, backward bounded) iff it is $F cofin$-bounded (resp. $F cofin$-forward bounded, $F cofin$-backward bounded).

We now give the fundamental notions of filter convergence and related topics in the cone metric space setting. Without loss of generality, we consider the symmetric case. Analogously it is possible to deal with the corresponding ‘backward’ and ‘forward’ concepts in the asymmetric case (see also [34]).

A net $( x λ ) λ ∈ Λ$ in a cone metric space R $(OF)$-converges to $x∈R$ (shortly, $(OF) lim λ x λ =x$) iff there exists an $(O)$-sequence $( σ p ) p$ in Y with ${λ∈Λ:ρ( x λ ,x)≤ σ p }∈F$ for each $p∈N$. A net $( x λ ) λ ∈ Λ$ in R is $(OF)$-Cauchy iff there is an $(O)$-sequence $( τ p ) p$ in Y such that for every $p∈N$ there is $F 0 ∈F$ with $ρ( x λ , x ξ )≤ τ p$ for each $λ,ξ∈ F 0$. Note that, since R is Dedekind complete, a net $( f λ ) λ$ in R is $(OF)$-convergent if and only if it is $(OF)$-Cauchy (see also [33, 48]).

Let Ξ be any nonempty set. We say that a family ${ ( x λ , ξ ) λ :ξ∈Ξ}$ in R $(OF)$-converges to $x ξ ∈R$ uniformly with respect to $ξ∈Ξ$ (shortly, $(UOF)$-converges to $x ξ$) as λ varies in Λ iff there is an $(O)$-sequence $( v p ) p$ in Y with

A family ${ ( x λ , ξ ) λ :ξ∈Ξ}$ $(ROF)$-converges to $x ξ ∈R$ (as λ varies in Λ) iff there exists an $(O)$-sequence $( σ p ) p$ in Y such that for each $p∈N$ and $ξ∈Ξ$ we get ${λ∈Λ:ρ( x λ , ξ , x ξ )≤ σ p }∈F$. By $(RO)$-convergence we will denote the $(RO F Λ )$-convergence. Observe that, when $R=Y=R$, $(ROF)$-convergence coincides with usual filter convergence (see also [8, 9, 33]).

Let $x∈X$. A net $f λ :X→R$, $λ∈Λ$, is said to be -exhaustive at x iff there is an $(O)$-sequence $( σ p ) p$ such that for any $p∈N$ there exist a neighborhood U of x and a set $F∈F$ such that for each $λ∈F$ and $z∈U$ we have $ρ( f λ (z), f λ (x))≤ σ p$.

A net $f λ :X→R$, $λ∈Λ$, is weakly -exhaustive at x iff there is an $(O)$-sequence $( σ p ) p$ such that for each $p∈N$ there is a neighborhood U of x such that for every $z∈U$ there is $F z ∈F$ with $ρ( f λ (z), f λ (x))≤ σ p$ whenever $λ∈ F z$.

We say that $f λ :X→R$, $λ∈Λ$, is (weakly) -exhaustive on X iff it is (weakly) -exhaustive at every $x∈X$ with respect to a single $(O)$-sequence, independent of $x∈X$.

Similarly as above it is possible to formulate the concepts of (weak) -forward (backward) exhaustiveness (see also [32]).

Of course the concepts of (weak, forward, backward) filter exhaustiveness can be given also analogously for sequences of functions, by taking $Λ=N$ with the usual order.

In the next section we will see that, in general, the notion of weak -exhaustiveness is strictly weaker than that of -exhaustiveness (for the case $R=Y=R$ see also [[30], Remark 2.8]).

## 3 The main results

We now give, in the context of filter convergence and lattice groups, a necessary and sufficient condition under which the limit of a pointwise convergent net $( f λ ) λ$ is (globally) continuous, extending [[31], Proposition 17] to nets of lattice group-valued functions.

Theorem 3.1 Under the same above notations and assumptions, let be a $(Λ)$-free filter of Λ, fix $x∈X$, and suppose that $f λ :X→R$, $λ∈Λ$, $(ROF)$-converges to $f:X→R$ on X with respect to a single $(O)$-sequence $( σ p ∗ ) p$ in Y. Then the following are equivalent:

1. (i)

$( f λ ) λ$ is weakly -exhaustive at x;

2. (ii)

f is continuous at x.

Proof (i) (ii) Let $( σ p ) p$ be an $(O)$-sequence in Y associated with weak -exhaustiveness of $( f λ ) λ$ at x, and pick $p∈N$. By hypothesis, there exists a neighborhood $U x$ of x, related with weak -exhaustiveness. Fix arbitrarily $z∈ U x$. There is a set $F 1 ∈F$ (depending on x and z) with $ρ( f λ (x), f λ (z))≤ σ p$ for all $λ∈ F 1$. Moreover, thanks to $(ROF)$-convergence with respect to the $(O)$-sequence $( σ p ∗ ) p$, there exists a set $F 2 ∈F$ (depending on x and z) with $ρ( f λ (z),f(z))≤ σ p ∗$ and $ρ( f λ (x),f(x))≤ σ p ∗$ whenever $λ∈ F 2$. Thus for every $λ∈ F 1 ∩ F 2$ we get

$ρ ( f ( x ) , f ( z ) ) ≤ρ ( f ( x ) , f λ ( x ) ) +ρ ( f λ ( x ) , f λ ( z ) ) +ρ ( f λ ( z ) , f ( z ) ) ≤2 σ p ∗ + σ p .$
1. (ii)

(i) Since f is continuous at x, there exists an $(O)$-sequence $( τ p ) p$ such that for each $p∈N$ there is a neighborhood $U x$ of x with

$ρ ( f ( x ) , f ( z ) ) ≤ τ p$
(1)

whenever $z∈ U x$.

By $(ROF)$-convergence of $( f λ ) λ$ to f on X with respect to the $(O)$-sequence $( σ p ∗ ) p$, there is a set $F ∗ ∈F$ with

$ρ ( f λ ( x ) , f ( x ) ) ≤ σ p ∗ andρ ( f λ ( z ) , f ( z ) ) ≤ σ p ∗$
(2)

for each $λ∈ F ∗$. From (1) and (2) we obtain

$ρ ( f λ ( x ) , f λ ( z ) ) ≤ρ ( f λ ( x ) , f ( x ) ) +ρ ( f λ ( z ) , f ( z ) ) +ρ ( f ( x ) , f ( z ) ) ≤2 σ p ∗ + τ p$
(3)

for every $λ∈ F ∗$. From (3) we get the existence of an $(O)$-sequence $( v p ) p$ with the property that for every $z∈ U x$ there is a set $F ∗ ∈F$ (depending on x and z) with $ρ( f λ (x), f λ (z))≤ v p$ whenever $λ∈ F ∗$. Thus the net $( f λ ) λ$ is weakly -exhaustive at x. This ends the proof. □

Analogously as Theorem 3.1, it is possible to prove the following.

Theorem 3.2 Under the same notations and assumptions as in Theorem  3.1, suppose that $f λ :X→R$, $λ∈Λ$, $(ROF)$-converges to $f:X→R$ on X with respect to a single $(O)$-sequence $( σ p ∗ ) p$ in Y. Then the following are equivalent:

1. (i)

$( f λ ) λ$ is weakly -exhaustive on X;

2. (ii)

f is globally continuous on X.

We now use Theorem 3.2 to show that, in general, -exhaustiveness is strictly stronger than weak -exhaustiveness.

Example 3.3 Let be any fixed free filter of , $X=[−1,1]$ be endowed with the usual distance, Σ be the σ-algebra of all Borel subsets of X, ν be the Lebesgue measure on X, and R be the space of all bounded ν-measurable functions on X, with identification up to ν-null sets. Note that order convergence in R coincides with almost everywhere convergence dominated by an element of R, which does not have a topological nature (see also [40]). For each real number a, let $a ̲$ the function which associates to every element $x∈X$ the constant a.

We consider the sequence $f n :X→R$, $n∈N$, defined as follows:

It is easy to see that $( f n ) n$ $(OF)$-converges pointwise to $0 ̲$ with respect to an $(O)$-sequence independent of $x∈X$ (for example, $σ p = 1 p ̲$, $p∈N$) and thus, by Theorem 3.2, $( f n ) n$ is weakly -exhaustive on $[−1,1]$. On the other hand, it is not hard to see that, if $( σ p ) p$ is any $(O)$-sequence in R, then there is a positive real number $M 0$ with $σ p (x)≤ M 0$ for every $p∈N$ and $x∈X$. In correspondence with $M 0$, for each set $F∈F$ there is an integer $n 0 ∈F$, $n 0 > M 0$. So we get $f n 0 ( 1 n 0 )= n ̲ 0$, and hence $f n 0 ( 1 n 0 )≰ σ 1$. From this it follows that $( f n ) n$ is not -exhaustive at 0. Moreover, it is not difficult to check that $( f n ) n$ is -backward exhaustive at 0, but not -forward exhaustive at 0.

The following two propositions extend [[49], Proposition 3.5] and will be useful in the sequel.

Proposition 3.4 Let , X, R be as in Theorem  3.2, $f λ :X→R$, $λ∈Λ$, be a function net, -exhaustive on X and $(ROF)$-convergent to $f∈ R X$ on X.

Then f is globally continuous on X, and the net $( f λ ) λ$ $(UOF)$-converges on every compact subset $C⊂X$ with respect to a single $(O)$-sequence, independent of C.

Proof Global continuity of f on X follows from Theorem 3.2.

Let now $( σ p ) p$, $( σ p ∗ ) p$, $( τ p ) p$ be three $(O)$-sequences in Y, related with -exhaustiveness of $( f λ ) λ$, $(ROF)$-convergence on X and global continuity of f on X respectively, let $C⊂X$ be any compact set, and choose arbitrarily $p∈N$ and $x∈C$. As $( f λ ) λ$ is -exhaustive at x and f is globally continuous, there exist $F x ∈F$ and an open neighborhood $U x$ of x, with

$ρ ( f λ ( z ) , f λ ( x ) ) ≤ σ p andρ ( f ( z ) , f ( x ) ) ≤ τ p$
(4)

for each $λ∈ F x$ and $z∈ U x$. By compactness of C there is a finite family ${ U x 1 , U x 2 ,…, U x k }$, with $x j ∈X$ for every $j∈[1,k]$, whose union contains C. Since $( f λ ) λ$ $(ROF)$-converges to f on X, in correspondence with $p∈N$ and $x 1 ,…, x k$ there is a set $F 0 ∈F$ with

(5)

Let $F:= F 0 ∩( ⋂ j = 1 k F x j )$: note that $F∈F$. Pick arbitrarily $z∈C$: there exists at least $j∈[1,k]$ with $z∈ U x j$. Then from (4) and (5) we get

$ρ ( f λ ( z ) , f ( z ) ) ≤ρ ( f λ ( z ) , f λ ( x j ) ) +ρ ( f λ ( x j ) , f ( x j ) ) +ρ ( f ( x j ) , f ( z ) ) ≤ σ p + σ p ∗ + τ p$

for each $λ∈F$. This ends the proof. □

Proposition 3.5 Let , X, R be as above. If $f λ :X→R$, $λ∈Λ$, is a net of functions, globally continuous with respect to a single $(O)$-sequence independent of λ and $(UOF)$-convergent to $f∈ R X$ on X, then f is globally continuous and $( f λ ) λ$ is -exhaustive on X.

Proof We begin with proving global continuity of f on X. Let $x∈X$, $( σ p ) p$ and $( τ p ) p$ be two $(O)$-sequences, related with $(UOF)$-convergence of $( f λ ) λ$ to f and global continuity of the $f λ$’s, respectively, and fix arbitrarily $x∈X$ and $p∈N$. By hypothesis there is a set $F∈F$ with $ρ( f λ (z),f(z))≤ σ p$ for each $λ∈F$ and $z∈X$. Fix $λ ¯ ∈F$. By global continuity of $f λ ¯$ there are a neighborhood U of x with $ρ( f λ ¯ (z), f λ ¯ (x))≤ τ p$ for each $z∈U$. Thus for such z’s we get

$ρ ( f ( z ) , f ( x ) ) ≤ ρ ( f ( z ) , f λ ¯ ( z ) ) + ρ ( f λ ¯ ( z ) , f λ ¯ ( x ) ) + ρ ( f λ ¯ ( x ) , f ( x ) ) ≤ 2 σ p + τ p ,$
(6)

namely global continuity of f with respect to the $(O)$-sequence $( v p ) p$, where $v p :=2 σ p + τ p$, $p∈N$.

We now prove -exhaustiveness of the net $( f λ ) λ$ on X. Choose arbitrarily $x∈X$. By global continuity of f with respect to $( v p ) p$, in correspondence with $p∈N$ and $x∈X$ there is a neighborhood $U x$ of x with $ρ(f(z),f(x))≤ v p$ whenever $z∈ U x$. By $(UOF)$-convergence of $( f λ ) λ$ to f on X with respect to $( σ p ) p$, there is $F ∗ ∈F$ with

(7)

From (6) and (7) we get

$ρ ( f λ ( z ) , f λ ( x ) ) ≤ρ ( f λ ( z ) , f ( z ) ) +ρ ( f λ ( x ) , f ( x ) ) +ρ ( f ( z ) , f ( x ) ) ≤2 σ p + v p$

for every $λ∈ F ∗$ and $z∈ U x$. This ends the proof. □

Remark 3.6 Proceeding analogously as above, it is possible to see that Theorems 3.1, 3.2 and Propositions 3.4, 3.5 hold even when the distance function ρ does not satisfy necessarily symmetric property, and $(ROF)$- ($(UOF)$-) convergence, (weak) -exhaustiveness and continuity are replaced by $(ROF)$- ($(UOF)$-) forward (backward) convergence, (weak) -forward (backward) exhaustiveness and forward (backward) continuity, respectively, under the hypothesis that the forward and backward convergences are equivalent.

Example 3.7 In general, the hypothesis that forward convergence implies backward convergence is essential. Indeed, let $X=[0,1]$ be endowed with the usual distance, $R=[0,1]×[0,1]$ be endowed with the following distance function:

Observe that $(R,d)$ is an asymmetric metric space (see also [[34], Example 5.10]). Let $f n :X→R$, $n∈N$, be defined by setting $f n (x)=( 1 n ,x)$, $n∈N$, $x∈[0,1]$. Note that $d((0,x),( 1 n ,x))= 1 n$ and $d(( 1 n ,x),(0,x))=1$. Thus, it is not difficult to check that for any free filter of the sequence $( f n ( x ) ) n$ -forward converges uniformly on $[0,1]$ to the function $f:[0,1]→R$ defined by $f(x)=(0,x)$, $x∈[0,1]$ and that the -forward limit is unique, while the -backward limit of $( f n ( x ) ) n$ does not exist in R for any $x∈[0,1]$. Moreover, since $d((0,x),(0,0))=d((0,0),(0,x))=1$ for each $x∈(0,1]$, we find that f is neither forward nor backward continuous at 0. Furthermore, as

$d ( f n ( x ) , f n ( 0 ) ) =d ( ( 1 n , x ) , ( 1 n , 0 ) ) =x=d ( ( 1 n , 0 ) , ( 1 n , x ) ) =d ( f n ( x ) , f n ( 0 ) )$

for every $x∈[0,1]$, it follows that for any free filter of the sequence $( f n ( x ) ) n$ is both -forward and -backward exhaustive at 0.

We now give some versions of Ascoli-type theorems in the context of lattice groups and filter exhaustive nets, extending earlier results proved in [30] and [32]. Note that in our context, since we deal with abstract structures which are not necessarily by a topology, it will be advisable to deal with suitable notions of ‘filter closedness’ and ‘filter compactness’ in relation with convergences, which are not necessarily generated by a Hausdorff topology. For example, note that in the space $L 0 ([0,1],Σ,ν)$ of all measurable functions on $[0,1]$ with respect to the σ-algebra Σ of all Borel subsets of $[0,1]$ and the Lebesgue measure ν, with identification up to ν-null sets, order convergence coincides with almost everywhere convergence, which does not have a topological nature. Moreover, there exist Dedekind complete vector lattices which do not have any Hausdorff compatible vector topology, for which every bounded monotone increasing sequence converges to its supremum (see for instance [40]).

Given a directed set Λ, a $(Λ)$-free filter of Λ, a topological space X, a cone metric space R and a nonempty set $Φ⊂ R X$, we say that Φ is $(ROF)$-compact (resp. $(cF)$-compact) iff every net $( f λ ) λ ∈ Λ$ in Φ admits a subnet $( f λ κ ) κ ∈ Λ$, $(ROF)$-convergent to an element $f∈Φ$ (resp. $(UOF)$-convergent to an element $f∈Φ$ on every compact subset $C⊂X$ with respect to a single $(O)$-sequence independent of C). We say that Φ is $(ROF)$-closed iff $f∈Φ$ whenever $( f λ ) λ$ is a net in Φ, $(ROF)$-convergent to $f∈ R X$. The $(ROF)$-closure of Φ is the set of the functions $f∈ R X$, having a net $( f λ ) λ$ in Φ, $(ROF)$-convergent to f. Analogously as above, it is possible to formulate the notions of $(cF)$-closedness and of $(cF)$-closure. Note that Φ is $(ROF)$-closed (resp. $(cF)$-closed) if and only if it coincides with its $(ROF)$-closure (resp. $(cF)$-closure).

When $R=R$, $(ROF)$-convergence coincides with -pointwise convergence, and hence we denote the related above concepts by -compactness, -closedness and -closure, respectively.

We now are in a position to give the following abstract Ascoli-type theorem.

Theorem 3.8 Under the same notations and hypotheses as above, if $Φ⊂Ψ⊂ R X$, where Φ is $(cF)$-closed and Ψ is $(ROF)$-compact, and

(H′) every $(ROF)$-convergent net $( h λ ) λ ∈ Λ$ in Φ has a subnet $( h λ ξ ) ξ ∈ Λ$, $(ROF)$-convergent (in $R X$) and -exhaustive on X,

then Φ is $(cF)$-compact.

Moreover, if Φ is $(cF)$-compact, then Φ satisfies condition (H′).

Proof We begin with the first part. Let $Φ⊂ R X$ be $(cF)$-closed, and $( f λ ) λ ∈ Λ$ be a net in Φ. Since $Φ⊂Ψ$ and Ψ is $(ROF)$-compact, $( f λ ) λ$ has a $(ROF)$-convergent subnet $( f λ κ ) κ ∈ Λ$. By condition (H′), this subnet has a sub-subnet $( f λ κ ζ ) ζ ∈ Λ$, $(ROF)$-convergent to a function $f∈ R X$ and -exhaustive on X. By Proposition 3.4, the net $( f λ κ ζ ) ζ$ -converges to f uniformly on every compact subset $C⊂X$, with respect to an $(O)$-sequence independent of C. Since Φ is $(cF)$-closed, then $f∈Φ$. Therefore, Φ is $(cF)$-compact.

We now turn to the last part. Let Φ be $(cF)$-compact and $( f λ ) λ ∈ Λ$ be a $(ROF)$-convergent net in Φ. Then it admits a subnet $( f λ κ ) κ ∈ Λ$, $(UOF)$-convergent to a function $f∈ R X$ on every compact subset $C⊂X$ with respect to an $(O)$-sequence, independent of C. By Proposition 3.5, we find that $( f λ κ ) κ ∈ Λ$ $(ROF)$-converges to f and is -exhaustive on every compact set C with respect to a single $(O)$-sequence, and hence, by arbitrariness of C, it $(ROF)$-converges to f and is -exhaustive on X. This ends the proof. □

Remark 3.9 Observe that, in general, condition (H′) is strictly weaker than equicontinuity (see also [[42], Remark 3.2.17]).

A consequence of Theorem 3.8 is the following (see also [32]).

Theorem 3.10 Let $(X, d X )$ and $(R, d R )$ be asymmetric metric spaces, such that $d X$ and $d R$ are real-valued distance functions, $y ¯$ be a fixed element of R, be any free filter of and suppose that

• (3.10.1) each subset of R, -closed and -forward bounded with respect to $y ¯$, is -compact.

• Let $Φ⊂ R X$ be such that

• (3.10.2) every sequence $( f n ) n$ in Φ, pointwise convergent in $R X$, has a $F cofin$-exhaustive subsequence;

• (3.10.3) every sequence $( f n ) n$ in Φ has a subsequence $( f n r ) r$, -pointwise forward bounded in R with respect to $y ¯$.

Then every sequence $( f n ) n$ in Φ admits a subsequence, uniformly convergent on every compact subset $C⊂X$ in the usual sense.

Proof Choose arbitrarily $x∈X$, set $Φ x :={f(x):f∈Φ}$, and let $Φ ¯ x$ be the closure of $Φ x$ in R. We claim that $Φ ¯ x$ is compact in R. Indeed, let $y∈ Φ ¯ x$. There exists a sequence $( y n ) n$ in $Φ x$, convergent to y with respect to $d R$. So, in correspondence with $ε=1$ there is a natural number $n 0 ∈N$ with $d R (y, y n )≤1$ whenever $n≥ n 0$. Moreover, there is a sequence $( f n ) n$ in Φ such that $f n (x)= y n$ for each $n∈N$. By (3.10.3), there exist a subsequence $( f n r ) r$ of $( f n ) n$, a positive real number $k x$ and a set $F x ∈F$ with $d R ( y ¯ , y n r )≤ k x$ whenever $r∈ F x$. Thus there exists $r 0 ∈N$ with

$d R ( y ¯ ,y)≤ d R ( y ¯ , y n r 0 )+ d R ( y n r 0 ,y)≤ k x +1,$

getting forward boundedness of $Φ ¯ x$ and hence -compactness of $Φ ¯ x$, thanks to the hypotheses. This implies that $Φ ¯ x$ is also compact: indeed, by proceeding analogously as in [[50], Proposition 2.4], it is possible to prove that every -convergent sequence in an asymmetric metric space $(X,d)$ has a subsequence, convergent in the usual sense. As $Φ ¯ x$ is compact in R, then, by virtue of the Tychonoff theorem, the set $∏ x ∈ X Φ ¯ x$ is compact in $R X$ with respect to the pointwise convergence. Since $Φ⊂ ∏ x ∈ X Φ ¯ x$, then we find that every sequence $( f n ) n$ in Φ has a subsequence $( f n k ) k$, pointwise convergent to a suitable function $h∈ R X$. From (3.10.2) and Theorem 3.8 used with $Y=R$, $Λ=N$ and $F= F cofin$, where the roles of Φ and Ψ are played by the $(cF)$-closure $Φ ¯$ of Φ and $∏ x ∈ X Φ ¯ x$ respectively, we get the assertion. □

Note that Theorem 3.10 extends [[30], Theorem 3.7] and [[32], Theorem 3.4]. Indeed, we get the following.

Corollary 3.11 Under the same hypotheses and notations as in Theorem  3.10, let be a P-filter of . Let $Φ⊂ R X$ satisfy (3.10.1), (3.10.2) and be such that

(3.11.1) every sequence in Φ, pointwise convergent in $R X$, has a -exhaustive subsequence.

Then every sequence $( f n ) n$ has a subsequence, uniformly convergent on X in the usual sense.

Proof Let $( f n ) n$ be any function sequence in Φ. Arguing analogously as in the proof of Theorem 3.10, we see that $( f n ) n$ has a subsequence $( f n k ) k$, pointwise convergent in the usual sense to a function $h∈ R X$. By (3.11.1), $( f n k ) k$ has a -forward exhaustive subsequence, say $( h n ) n$. Since is a P-filter, proceeding analogously as in [[30], Lemma 3.6], it is possible to see that $( h n ) n$ has a $F cofin$-forward exhaustive subsequence. The assertion follows from Theorem 3.10. □

Example 3.12 In general, in Theorem 3.10, the condition (3.10.1) cannot be dropped. Indeed, let X, $(R,d)$ and $f n :X→R$, $n∈N$, be as in Example 3.7. Note that $(R,d)$ is bounded and closed (with respect to itself), and also -closed and -bounded for any free filter of . Let $A={ x n :=(0, 1 n ):n∈N}$: since $d((0, 1 n ),(0,b))=1$ whenever $b≠ 1 n$ and $d((0, 1 n ),(a,b))≥a$ for every $a∈(0,1]$, $b∈[0,1]$ and $n∈N$, it is not difficult to see that, for each free filter of , every subsequence $( x n r ) r$ of $( x n ) n$ does not admit -forward limit as $r→+∞$. From this it follows that $(R,d)$ is not -forward compact and hence it does not fulfil condition (3.10.1). Let now $Φ:{ f n :n∈N}$: it is not hard to check that the conditions (3.10.2) and (3.10.3) are satisfied. Note that, as seen in Example 3.7, the sequence $( f n ) n$ does not have any subsequence, uniformly (backward) convergent in the usual sense, and so the thesis of Theorem 3.10 is false.

We now give some versions of abstract Ascoli-type theorems with respect to Lipschitz-type metrics. For a related literature, we refer to [35] and the bibliography therein.

Let $(X,d)$ be a metric space endowed with a real-valued distance function, R be a Dedekind complete vector lattice, $Y=R$ endowed with the absolute value, and let us add to R an extra element +∞, satisfying the properties analogous to those of the element +∞ of the extended real line. We say that $f:X→R$ is Lipschitz iff there is a positive element $M∈R$ with $|f( x 1 )−f( x 2 )|≤d( x 1 , x 2 )M$ whenever $x 1 , x 2 ∈X$, and in this case we set

$Π(f):=⋁ { | f ( x 1 ) − f ( x 2 ) | d ( x 1 , x 2 ) : x 1 , x 2 ∈ X : x 1 ≠ x 2 } .$
(8)

If $f:X→R$ is not Lipschitz, then we put $Π(f):=+∞$. Note that, even if X is a compact metric space, $R=R$ and $f:X→R$ is continuous, it may happen that $Π(f)=+∞$: indeed, it is enough to take $f(x)= x 1 / 2$, $x∈[0,1]$.

We now fix a point $x 0 ∈X$ and consider the following extended metric:

$d L ( f 1 , f 2 ):= | f 1 ( x 0 ) − f 2 ( x 0 ) | ∨Π( f 1 − f 2 ), f 1 , f 2 :X→R.$
(9)

Given a directed set $(Λ,≥)$ and a $(Λ)$-free filter of Λ, we say that

$(F) lim λ ∈ Λ d L ( f λ ,f)=0or(F d L ) lim λ ∈ Λ f λ =f$

iff there is an $(O)$-sequence $( σ p ) p$, with

In this case, we say that the net $( f λ ) λ$ $(F d L )$-converges to f. The net $( f λ ) λ$ is $(F d L )$-Cauchy iff there is an $(O)$-sequence $( σ p ) p$ in R with the property that for every $p∈N$ there is $F∈F$ with $d L ( f ξ , f ζ )≤ σ p$ whenever $ξ,ζ∈F$.

It is easy to check that every $(F d L )$-convergent net is $(F d L )$-Cauchy. We will prove also the converse implication. To this aim, we first give the following extension of [[35], Proposition 3.5] to the setting of filters and vector lattices.

Proposition 3.13 Let $f λ :X→R$, $λ∈Λ$, be a function net, $(F d L )$-convergent to f, and $x 0$ be related with $d L$. Then, for every $k>0$, $( f λ ) λ$ $(UOF)$-converges to f on the set $S( x 0 ,k):={x∈X:d(x, x 0 ).

Proof By hypothesis, there is an $(O)$-sequence $( σ p ) p$ such that for every $p∈N$ there is a set $F∈F$ with

$| f λ ( x 0 ) − f ( x 0 ) | ≤ σ p and | ( f λ − f ) ( x ) − ( f λ − f ) ( x 0 ) | ≤d(x, x 0 ) σ p$
(10)

for every $λ∈F$ and $x∈X$. From (10) we get

$| f λ ( x ) − f ( x ) | ≤(1+k) σ p$

whenever $λ∈F$ and $x∈S( x 0 ,k)$. This ends the proof. □

Example 3.14 Observe that, in general, $(F d L )$-convergence is strictly stronger than $(UOF)$-convergence on compact sets. Indeed, let $(X,d)$ be a compact metric space, fix a point $x∈X$, and suppose that there exists at least a sequence $( x n ) n$ in X, convergent to x with respect to d in the usual sense. Let R be any Dedekind complete vector lattice, u be a strictly positive element of R, and put $f n (t)=d(t, x n )u$, $f(t)=d(t,x)u$ for every $n∈N$ and $t∈X$. It is not difficult to check that $| f n (t)−f(t)|=|d(t, x n )u−d(t,x)u|≤d(x, x n )u$ for any $n∈N$ and $t∈X$. From this it follows that the sequence $( f n ) n$ $(UOF)$-converges on X for every free filter of . On the other hand, we get

$Π ( f n − f ) ≥ | ( f n − f ) ( x n ) − ( f n − f ) ( x ) | d ( x n , x ) = | d ( x n , x n ) u − d ( x , x n ) u − d ( x n , x ) u + d ( x , x ) u | d ( x n , x ) = 2 u$

for every $n∈N$. This implies that, for any free filter of , the sequence $( Π ( f n − f ) ) n$ does not converge to 0. Otherwise there exists an $(O)$-sequence $( σ p ) p$ in R such that ${n∈N:Π( f n −f)≤ σ p }∈F$ for every $p∈N$, and so for each $p∈N$ there is $n p ∈N$ with $2u≤Π( f n p −f)≤ σ p$. By arbitrariness of p it follows that $u=0$, but this is impossible. Thus $( Π ( f n − f ) ) n$ does not $(OF)$-converge to 0, and hence $( f n ) n$ is not $(F d L )$-convergent to f (see also [[35], Example 3.1]).

As a consequence of Proposition 3.13, we prove the following completeness result, which extends [[35], Proposition 3.3].

Proposition 3.15 Under the same above notations and hypotheses, let $f λ :X→R$, $λ∈Λ$, be an $(F d L )$-Cauchy net of functions, globally continuous with respect to a single $(O)$-sequence. Then $( f λ ) λ$ $(F d L )$-converges to a globally continuous function $f:X→R$.

Proof Let $f λ :X→R$, $λ∈Λ$, be a $(F d L )$-Cauchy net, and $x 0$ be related with $d L$. Then the net $( f λ ( x 0 ) ) λ$ is $(OF)$-Cauchy, and so it is $(OF)$-convergent, since R is Dedekind complete (see also [48]). Moreover, for every $λ,ξ∈Λ$ and $x∈X$ we get

$| f λ ( x ) − f ξ ( x ) | ≤ | f λ ( x 0 ) − f ξ ( x 0 ) | + d ( x , x 0 ) Π ( f λ − f ξ ) ≤ | f λ ( x 0 ) − f ξ ( x 0 ) | + d ( x , x 0 ) d L ( f λ , f ξ ) .$
(11)

From (11), the hypotheses and since the net $( f λ ( x 0 ) ) λ$ is $(OF)$-Cauchy, it follows that for every $x∈X$ the net $( f λ ( x ) ) λ$ is $(OF)$-Cauchy, and so $(OF)$-convergent. For each $x∈X$, set $f(x):=(OF) lim λ f λ (x)$.

We now prove that $(F) lim λ Π( f λ −f)=0$. Choose arbitrarily $p∈N$. Since by hypothesis $( f λ ) λ$ is $d L$-Cauchy, there is $F∈F$ with $d L ( f λ , f ξ )≤ σ p$, and hence

(12)

From (12), since $f λ (x)−f(x)=(OF) lim ξ ( f λ (x)− f ξ (x))$ for every $x∈X$ and $λ∈Λ$, it follows that

(13)

From Propositions 3.5, 3.13 and (13) we get global continuity of f. This ends the proof. □

The next step is to give an Ascoli-type theorem involving $d L$. We say that a net $f λ :X→R$, $λ∈Λ$, is -finitely $d L$-bounded iff there exists a finite number q of globally continuous functions $h 1 ,…, h q ∈ R X$, of elements $r 1 ,…, r q ∈R$, of sets $E 1 ,…, E q$ with $Λ= ⋃ j = 1 q E j$, and a set $F∈F$ such that $d L ( f λ , h j )≤ r j$ for every $j∈[1,q]$ and whenever $λ∈F∩ E j$.

Theorem 3.16 Let X be a metric space. If $Φ⊂Ψ⊂ R X$, where Φ is $(cF)$-closed and Ψ is $(ROF)$-compact, and if we assume that

(3.16.1) every $(ROF)$-convergent net in Φ is -finitely $d L$-bounded,

then Φ is $(cF)$-compact.

Proof Choose arbitrarily a net $( f λ ) λ ∈ Λ$. By $(ROF)$-compactness of Ψ, $( f λ ) λ ∈ Λ$ has a subnet $( f λ ζ ) ζ ∈ Λ$, $(ROF)$-convergent to a function $f∈ R X$. By (H′), $( f λ ζ ) ζ ∈ Λ$ is -finitely $d L$-bounded. Pick arbitrarily $x∈X$ and let q, F and $h j$, $r j$, $E j$, $j∈[1,q]$, be according to -finite $d L$-boundedness of $( f λ ζ ) ζ$. Since the $h j$’s are globally continuous, we find an $(O)$-sequence $( τ p ) p$ such that for each $p∈N$ there is a neighborhood $U x$ of x with

(14)

If $ζ∈F∩ E j$ and $z∈ U x$, then from (14) we get

$| f λ ζ ( z ) − f λ ζ ( x ) | = | ( f λ ζ ( z ) − h j ( z ) ) + ( h j ( z ) − h j ( x ) ) + ( h j ( x ) − f λ ζ ( x ) ) | ≤ | h j ( z ) − h j ( x ) | + | ( f λ ζ − h j ) ( z ) − ( f λ ζ − h j ) ( x ) | ≤ τ p + d ( z , x ) Π ( f λ ζ − h j ) ≤ τ p + d ( z , x ) d L ( f λ ζ , h j ) ≤ τ p + d ( z , x ) r j ,$
(15)

getting -exhaustiveness of $( f λ ζ ) ζ ∈ Λ$. By Proposition 3.4, the net $( f λ ζ ) ζ ∈ Λ$ $(UOF)$- converges to f on every compact subset $C⊂X$ with respect to a single $(O)$-sequence, independent of C. This ends the proof. □

When $R=R$ and $Λ=N$, it is possible to prove the following extension of [[35], Theorem 5.1].

Theorem 3.17 Let be any free filter of , X be a separable metric space, $Φ⊂ R X$ be $(c F cofin )$-closed, and such that every sequence $f n :X→R$, $n∈N$, in Φ is -finitely $d L$-bounded. Then Φ is $(c F cofin )$-compact.

Proof Let $( f n ) n$ be an -finitely $d L$-bounded sequence, $x 0$ be related to $d L$, and F, q, $h j$, $r j$, $E j$, $j∈[1,q]$, be associated with -finite $d L$-boundedness of $( f n ) n$. By arguing analogously as in the proof of Theorem 3.16, formula (15), it is possible to show that the sequence $f n$, $n∈N$, satisfies condition (3.10.1) with respect to the filter $F cofin$. To prove (3.10.2), observe that

(16)

From (16) it follows that

$| f n ( x ) | ≤ ⋁ j ∈ [ 1 , q ] | h j ( x ) | + r j +d(x, x 0 ) ⋁ j ∈ [ 1 , q ] r j$

for each $n∈F$ and $x∈X$. Thus the sequence $f n$, $n∈F$, satisfies (3.10.2) with respect to $F cofin$. By Theorem 3.10, the sequence $f n$, $n∈F$, and so even the sequence $f n$, $n∈N$, has a subsequence, uniformly convergent to f on every compact subset $C⊂X$ in the usual sense. By $(c F cofin )$-closedness of Φ, it follows that $f∈Φ$. This ends the proof. □

Example 3.18 Observe that, in Theorem 3.17, in general the hypothesis of -finite $d L$-boundedness cannot be dropped. Indeed, let $f n :[−1,1]→R$, $n∈N$, be a function sequence, such that $f n ([−1,− 1 n ])= f n ([ 1 n ,1])=0$, $f n (0)=n$ and $f n$ is linear in $[− 1 n ,0]$ and in $[0, 1 n ]$ for each $n∈N$. It is not difficult to check that for every free filter of , for any $F∈F$, for each $q∈N$ and for any choice of real numbers $r 1 ,…, r q$, of functions $h 1 ,…, h q$, continuous on $[−1,1]$, and of sets $E 1 ,…, E q$ with $⋃ j = 1 q E j =N$, there exists at least an index $j ¯ ∈[1,q]$ such that $F∩ E j ¯$ is infinite: otherwise F should be finite, but this is impossible, since is a free filter of . Thus there is a sufficiently large integer $n ¯ ∈F∩ E j ¯$, with

$Π( f n ¯ − h j ¯ )=sup { | ( f n ¯ − h j ¯ ) ( x 1 ) − ( f n ¯ − h j ¯ ) ( x 2 ) | | x 1 − x 2 | : x 1 , x 2 ∈ [ − 1 , 1 ] : x 1 ≠ x 2 } > r j ¯ .$

Thus, for any free filter of , the sequence $( f n ) n$ is not -finitely $d L$-bounded. Furthermore, it is not hard to see that the sequence $( f n ) n$ has no subsequence convergent uniformly on $[−1,1]$, and so the set $Φ:={ f n :n∈N}$ is $(c F cofin )$-closed, but not $(c F cofin )$-compact (see also [[35], Example 5.3]).

Remark 3.19 The results here obtained hold, even if Y is a weakly σ-distributive lattice group endowed with $(D)$-convergence (see for instance [51]).

Open problems

1. (a)

Prove similar results on (weak) filter exhaustiveness in other abstract structures.

2. (b)

Find further versions of Ascoli-type theorems, by considering different kinds of convergences in various kinds of abstract spaces.

## 4 Conclusion

We have first given, for cone metric space function nets, a characterization of continuity of the limit function in terms of filter weak exhaustiveness and some relation between filter exhaustiveness and uniform convergence (on compact sets). These relations have been used in proving our main abstract Ascoli-type theorem. Indeed, the condition on existence of a (filter pointwise convergent with respect to a single order sequence and) filter exhaustive subnet is necessary and sufficient for compactness with respect to the uniform convergence of compact sets, given the compactness with respect to the pointwise convergence. In the classical case, it is possible to treat these subjects by dealing with the compact-open topology (which corresponds to convergence uniformly on compacta; see also [41]). But in a general Dedekind complete lattice group it may happen the non-existence of Hausdorff topologies ‘compatible’ with the order (see also [40]), and we need an approach different from the classical one, by considering concepts like closedness and compactness by means of nets (or sequences) of functions rather than by topologies. In Theorem 3.10 and Corollary 3.11, we have extended earlier results proved in [32] and [30], including both the symmetric and the asymmetric case.

Furthermore, we have dealt with Lipschitz-type metrics and considered also extended distance-type functions, which are related with not necessarily Lipschitz functions. The main Ascoli-type results given in [35] have been extended in the lattice group setting and in the context of filter convergence and exhaustiveness (and also a concept similar to ‘total boundedness’ has been presented in terms of filters), giving some compactness result again with respect to uniform convergence on compact sets. We have given also some examples to support the results proved in our setting.

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## Acknowledgements

Our thanks to the referees for their remarks, which improved the exposition of the paper.

## Author information

Authors

### Corresponding author

Correspondence to Xenofon Dimitriou.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Boccuto, A., Dimitriou, X. Ascoli-type theorems in the cone metric space setting. J Inequal Appl 2014, 420 (2014). https://doi.org/10.1186/1029-242X-2014-420

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### Keywords

• lattice group
• cone metric space
• (free) filter
• global continuity
• filter convergence
• (weak) filter exhaustiveness
• Ascoli theorem
• Lipschitz metric