Topological Properties of Quasilinear Spaces and Set-Valued Maps

Topological Properties of Quasilinear Spaces and Set-Valued Maps

Sanjay Kumar Suman

        Department of Mathematics, Government Degree College, Bagaha, West Champaran, B.R.A. Bihar University, Muzaffarpur, Bihar, India.

DOI: https://doi.org/10.51244/IJRSI.2025.120600173

Received: 23 June 2025; Accepted: 26 June 2025; Published: 23 July 2025

ABSTRACT

This paper presents a comprehensive study on the topological properties within the framework of quasilinear spaces, particularly focusing on their application to set-valued maps. We generalize the established concept of normed quasilinear spaces, primarily as introduced by Aseev in 1986, to the more encompassing notion of topological quasilinear spaces1. Our investigation demonstrates that if S is an arbitrary topological space and Z is a metrizable topological vector space, then the space of minimal semi-continuous compact-valued maps, denoted as M(S,Y), can be mapped into a linear space. This significantly extends previous results where S was restricted to being a Baire space.

A key contribution of this work is the detailed analysis of the topological vector space M_s(S, Z), defined by the topology of strong uniform convergence on compact subsets of S. We establish several fundamental properties of this space, including its completeness when S is a locally compact Hausdorff space and Z is complete. Furthermore, we prove that M_s(S, Z) is metrizable if and only if S is hemicompact. The paper also undertakes a comparative study of the topologies τ_s and τ_k, demonstrating that τ_s is finer than τ_k, with equality holding when S is locally compact. Through rigorous mathematical illustrations and examples, we show that the principle of localization, typically valid for topological vector spaces, may not hold universally for all topologically quasilinear spaces, particularly in the context of singular elements. These findings collectively contribute fresh insights into the topology of quasilinear spaces, multivalued mappings, and set-valued analysis.

Keywords: Quasilinear Spaces, Set-Valued Maps, Topological Properties, Topological Vector Spaces, Metrizability, Completeness, Uniform Convergence, Hausdorff Space, Hemicompact Space, Minkowski Operations, Set-Valued Analysis, Aseev.

INTRODUCTION

The concept of Quasilinear Spaces introduced by Aseev encompassed both the traditional linear spaces & subset of non linear spaces and its multilevel mappings. This he applies to the quasilinear  space for its linear functional analysis by introducing the concept of  quasilinear operators & functional. From here, he proceeds further to exhibit some outcomes which in linear functional analysis context happens to be the quasilinear equivalent of the functional definition & theorem whereas in Banach Spaces context they were differential calculus. Aseev’s this path breaking work has motivated many, and this paper, to present new results on fuzzy quasilinear spaces, multileveled mappings & set-valued analysis.

The most significant example of a quasilinear space is K_c(E) when K_c(E) is a collection consisting of all convex compact subset of  E, and where E is a normed space. The study of this class includes convex and interval analysis since intervals turns out to be an ideal tools in the context of solving the global optimization problems as well as in the context of being an addition to the conventional techniques.

Intervals being a set with infinite number, possesses infinite amount of information i.e. worldwide information. Further, the set differential equations theory also requires the study of  K_C(E). There are various different methods including Markow method for introducing and for dealing with the quasilinear spaces. However, this paper but feels Aseev’s best suited to provide the foundation and tools. Also, the Aseev’s method provides higher dimension of set-valued algebra & analysis through ordering relation. Here in this paper the issues related to the problem of uncertainty and to sensitivity necessarily belongs to set-valued analysis. The origin of set-valued analysis traced back to 19th century, developed primarily by  Cauchy, Riemann and Weierstrass, has gathered much interest these days. Aseev’s early research in this aspect is also worthy of mention here. The main conclusions of this paper show that from S, an arbitrary topological space, the collection of minimal semi-continuous compact value maps into Z a metrizable topological vector space with Z being a vector space, this generalize the former results where S necessarily were Baire Space. The definitions of the algebraic operation, there, depended basically on the fact that any minimal upper semi-continuous compact valued map from  S, a Baire space,  to Y  where Y is a metric space is point-valued on a dense G-δ set.  Since this does not hold for generalized topological space S, new techniques have been considered in this paper. The metric characterization of  the quasi-minimal upper semi-continuous compact valued maps being the key.

In this paper, until and unless defined otherwise, S and Y continue to denote general topological spaces and Z continues to be a topological vector space over a field K, wherein K is either the field of real numbers, R or C a field of complex numbers. For each s ∈ S, Vx  still denotes the collection of an  open neighborhoods of s in S. whereas the closure of A ⊆ S still remain to be denoted by A,  such that Int(A)  is the interior of A.  In case Y is a metric space, it is denoted by B(y, ϵ). The open ball with center at y ∈ Y of radius ϵ > 0, where B(y, ϵ) denotes closure of B(y, ϵ).

Wherever it does so necessarily highlight the specific metric d on Y, the ball with center at y ∈ Y of radius ϵ is represented by Bd(y, ϵ). The collection consisting of all the subsets of Y is denoted here as P (Y), whereas   C(Y) denotes it as consisting of all closed and non-empty subsets of  Y. The collections consisting of all nonempty compact subsets of Y is denoted here as K(Y).

There following the establishment of normed quasilinear spaces and the bounded quasilinear operators in the already established norms, and by introducing some new results, this paper is able to some significant contributions towards the enhancement of the quasilinear functional analysis.

Algebraic Operations and Linear Structures of Set-Valued Maps:

For the sum of  subsets A and B of a topological vector space  Z,  Minkowski  defines as follow
A ⊕ B = {a + b : a ∈ A, b ∈ B},
and  the Minkowski product of A ⊆ Z and a scalar α is
α ⊙ A = {αa : a ∈ A}.                                                                                       (1)
owing  to the continuity of the algebraic operations on Z, K(Z) is closed under the Minkowski operations. That is,

⊕ : K(Z) × K(Z) → K(Z)	(2)
And
⊙ : K × K(Z) → K(Z).	(3)

Here, though. K(Z) isn’t a vector space with respect to (2) and (3). But naturally, although {0} ∈ K(Z) is an identity for (2) and the associativity and commutativity axioms are true, a general member in K(Z) doesn’t have an additive inverse. In fact,  A ∈ K(Z) have  additive inverse if and only if A is a singleton.  Furthermore, scalar multiplication (3) isn’t distributive over addition in K.

Using operations (2) and (3) in a point-wise manner to employ mappings  f, g : S → Z and α ∈ K, one has:

f ⊕ g : S ∋ s → f (s) ⊕ g(s) ∈ K(Z),	(4)
And
α ⊙ f : S ∋ s → α ⊙ f (s) ∈ K(Z)	(5)

are usco. Also, since Minkowski operations do not meet the axioms of linear space on K(Z), it is clear that the set of all usco maps form S into Z is not a linear space under (4) and (5). By proceeding with this, we can observe that an usco map  f : S ⇒ Z have an additive inverse if and only if  f  is point-valued at each s ∈ S.  This being the case it follows that   f   is a continuous function from S into Z. Thus, neither M(S, Z) nor Q(S, Z) is, in general, a linear space according to (4) and (5). Actually, M(S, Z) is not even closed under Minkowski addition
Example   Consider the musco maps f , g : R ⇒ R be defined as

                                   (6)

And,

                                                                                                                                                                                                                                                (7)

Therefore, Minkiwski sum of f ang g will be

                                           (9)

Now, it is clear that f ⊕ g is usco, but not minimal.

M (S, Z) As A Linear Space:

Assuming  Z to be metrizable.   M(S, Z) is not a linear space with and is also not closed under Minkowski addition and hence  also not even a quasilinear space. However, the pointwise addition in (4) can be used to define the sum of two musco maps in the following way: for f , g ∈ M(S, Z), define the sum f + g of f and g as
f + g = ⟨f ⊕ g⟩,

where ⟨·⟩ : Q(S, Z) → M(S, Z) is the map defined  and  f ⊕ g ∈ Q(S, Z) since  f and g, being musco,  a quasi-minimal usco. Therefore f + g  defined for all f , g ∈ M(S, Z). Easily it can be seen  that M(S, Z) is closed under Minkowski scalar multiplication (5). Therefore we defined the scalar product of α ∈ K and f ∈ M(S, Z) to be the Minkowski product of α and f .

That means,

αf = α ⊙ f .                                                                               (10)

Theorem 1: M(S,Z) is a linear space

Proof: From the above operations. Commutativity of the  addition, the existence of  the additive identity, distributivity of the  scalar multiplication over addition in M(S, Z), the compatibility of the scalar multiplication with field multiplication and also the identity element for scalar multiplication is the multiplicative identity in K all follow immediately, We  only checked  the rest of the axioms of the  linear space.
To find that the  addition is associative, assume  f , g, h ∈ M(S, Z). By now, it is clear that
f + (g + h) ⊆ f ⊕ (g + h) ⊆ f ⊕ (g ⊕ h) and
(f + g) + h ⊆ (f + g) ⊕ h ⊆ (f ⊕ g) ⊕ h.
Since f ⊕ (g ⊕ h) = (f ⊕ g) ⊕ h  it follows from the quasi-minimality of (f ⊕ g) ⊕ h that
f + (g + h) = (f + g) + h.
For f ∈ M(S, Z), let –f = (-1)f . Then by the definition (10) of addition in M(S, Z) it follows that f + (–f ) ⊆ f ⊕ (–f )
and since 0 ∈ f ⊕ (–f )(s) for every s ∈ S, it follows that f + (–f )(s) = ⟨f ⊕ (–f )⟩ (s) = {0}.
For α, β ∈ K and f ∈ M(S, Z), it follows  that
(α + β)f ⊆ (αf ) ⊕ (βf ).
But αf + βf ⊆ (αf ) ⊕ (βf ) by (12) so that αf + βf = ⟨(αf ) ⊕ (βf )⟩ = (α + β)f .
It is notable that the algebraic operations on M(S, Z) are natural in at least two ways. At first, in terms of the quasilinear space Q(S, Z), the linear space M(S, Z) may be viewed as a quotient space with respect to the quasilinear subspace
Q₀(S, Z) = {f ∈ Q(S, Z) : 0 ∈ f (s), s ∈ S}
of Q(S, Z). Indeed, if f , g ∈ Q(S, Z) then f ⊕ (-1 ⊙ g) ∈ Q0(S, Z) if and only if ⟨f ⟩ = ⟨g⟩. Therefore each f ∈ M(S, Z) may be viewed as an equivalence class of quasi-minimal usco maps given by
f + Q0(S, Z) = {g ∈ Q(S, Z) : f – g ∈ Q0(S, Z)} = {g ∈ Q(S, Z) : ⟨g⟩ = f }.
Then at second place, the algebraic operations of the M(S, Z) extend the usual point-wise operations on the set C(S, Z) of continuous functions from S into Z, in the way that the natural inclusion of C(S, Z) in M(S, Z) defines an injective linear transformation.

Topologies of Uniform Convergence on Compact Sets for Set-Valued Maps:

Consider Z be the metrizable topological vector space having translation invariant metric d_Z. The topology of uniform convergence on compact sets (τ_k) on M(S, Z) is defined in tandem with  Hausdorff metric H on K(Z),  is defined by the setting

H(K, L) = max{max{dZ (y, L) : y ∈ K}, max{dZ (z, K) : z ∈ L}}	(11)
for all K, L ∈ K(Z). Here d(z, K) = min{dZ (z, y) : y ∈ K} for all K ∈ K(Z). For f ∈ M(S, Z), ϵ > 0 and K ∈ K(S), let
W(f , K, ϵ) = {g ∈ M(S, Z) : H(f (s), g(s)) < ϵ, s ∈ K}.	(12)

It’s collection i.e. {W(f , K, ϵ) : f ∈ M(S, Z), K ∈ K(S), ϵ > 0} is the basis for the  “τ_k”  on  S. Following the previous operation we denote by M_k(S, Z) the set of musco maps from S into Z  being equipped with this topology. Based on  the above theorem  M_k(S, R) is a locally convex linear topological space whenever S is locally compact. Vice versa, if S  be a first countable, regular Baire space, then S is locally compact if and only if addition in M_k(S, R) is continuous,
While M_k(S, Z) is, in general, not a topological vector space, the collection B₀ = {W(0, K, ϵ) : K ∈ K(S), ϵ > 0} is a basis at 0 ∈ M(S, Z) in terms with  a vector space topology τ_s on M(S, Z). We call this topology τ_s the topology on strong uniform convergence on compact subsets of S, and denote M(S, Z) being equipped with this topology by M_s(S, Z). As shown, here we invigilate some of the properties of M_s(S, Z).

Properties of The Topological Vector Space M_s (S, Z):

Verifying our statement  that B₀ is a basis at 0 for a vector space topology on M(S, Z).
Theorem 2 : There is a Hausdorff vector space topology τ_s on M(S, Z) so that B₀ is a basis for the τ_s-neighbourhood filter at 0 ∈ M(S, Z).
Proof. We prove that V₀  the filter generated by the collection B₀ satisfies the conditions .. To this end, fix K ∈ K(S), ϵ > 0 and α ∈ K \ {0}.

Clearly 0 ∈ W(0, K, ϵ). Select f , g ∈ W(0, K, ϵ/2 ). If s ∈ K, y ∈ f (s) and            z ∈ g(s) then, owing to the translation invariance of d_Z, it follows that d_Z (y+z, 0) ≤ d_Z (y, 0)+d_Z (z, 0) < ϵ. Therefore f +g(s) ⊆ f ⊕g(s) ⊆ B(0, ϵ) such that H({0}, f + g(s)) < ϵ for all s ∈ K. Hence f + g ∈ W(0, K, ϵ) so that W(0, K, ϵ/ 2 ) + W(0, K, ϵ 2 ) ⊆ W(0, K, ϵ).
Accordingly there exists ϵ_α > 0 such that B(0, ϵ_α) ⊆ αB(0, ϵ). If f ∈ W(0, K, ϵ_α), then f (s) ⊆ B(0, ϵα) ⊆ αB(0, ϵ) so that (1/α) f (s) ⊆ B(0, ϵ) for every s ∈ K. Hence H({0}, (1/α) f (s)) < ϵ for each s ∈ K, thus (1/α) f ∈ W(0, K, ϵ). Hence
f ∈ αW(0, K, ϵ) so that W(0, K, ϵ_α) ⊆ αW(0, K, ϵ). If h ∈ M(S, Z) then, since K is compact, h(K) is compact. As such,  there exists a constant C > 0 so that λh(s) ⊆ λh(K) ⊆ B(0, ϵ) for all λ ∈ K so that |λ| < C and every s ∈ K. Thus H({0}, λh(s)) < ϵ for each s ∈ K so that λh ∈ W(0, K, ϵ) for each such . λ. That is, W(0, K, ϵ) is absorbing. It follows from  previous operations that there exists ϵ′ > 0 so that B(0, ϵ) contains the balanced hull of B(0, ϵ′). That is, {αy : y ∈ B(0, ϵ′), |α| ≤ 1} ⊆ B(0, ϵ). If f ∈ W(0, K, ϵ′) then f (s) ⊆ B(0, ϵ′) for every s ∈ K. Therefore αf (s) is contained in the balanced hull of B(0, ϵ′), and hence in B(0, ϵ), for every s ∈ S and α ∈ K so that |α| ≤ 1.  As a result αf ∈ W(0, K, ϵ) whenever f ∈
W(0, K, ϵ′) and |α| ≤ 1 so that W(0, K, ϵ) contains the balanced hull of W(0, K, ϵ′).  Hence every element of V₀ contains an element of V₀ that is balanced. Hence all the conditions in the operation  are satisfied so that V₀ is the neighbourhood filter at 0 ∈ M(S, Z) for a vector space topology τ_s on M(S, Z

Since

it follows that τ_s is Hausdorff.

It is note worthy that the topology τ_k on D(S, Z) depends on a particular metric d_Z on Z. It is also notable that the topology τ_k on D(S, Z) depends on the particular metric d_Z on Z.  That is, if d₁ and d₂ are metrics on Z which is compatible with the topology of Z, then the topology of uniform convergence on compact subsets of S generated by d₁ may differ from the one derived by the means of  d₂. Holá showed that if S is a regular, first countable and non discrete, then two compatible metrics on Z generate the same topology of uniform convergence on compact subsets of S on D(S, Z) if and only if the metrics are uniformly equivalent.

Theorem 3 :  If Z is locally convex, then M_s(S, Z) is a locally convex space.
Proof. Consider K ∈ K(S), ϵ > 0, f , g ∈ W(0, K, ϵ) and α ∈ [0, 1]. Then f (s), g(s) ⊆ B(0, ϵ) for all s ∈ K. If y ∈ αf +(1-α)g(s) for some s ∈ K, then there exists z ∈ f (s) and w ∈ g(s) so that y = αz+(1-α)w. Hence d_Z(y, 0) ≤ d_Z((1-α)w, 0)+d_Z(αz, 0). But we may assume that the metric d_Z on Z satisfies d_Z(αv, 0) ≤ |α|d_Z(v, 0) for all v ∈ Z and α ∈ K with |α| ≤ 1. Hence d_Z(y, 0) < ϵ. Since this holds for all s ∈ K and y ∈ αf + (1 – α)g(s) it follows that αf + (1 – α)g ∈ W(0, K, ϵ). Thus M_s(S, Z) is a locally convex linear topological space.

Completeness and Metrizability Of M_s (S, Z):

We here, consider issues related to completeness and metrizability. We show that M_s(S, Z) is complete, with respect to the natural uniformity induced by its vector space topology, whenever S is a locally compact Hausdorff space and Z is complete.
Furthermore, M_s(S, Z) is metrizable if and only if S is hemicompact. Recall that S is hemicompact if there exists a countable subset K₀ of K(S) so that every K ∈ K(S) is contained in a member of K₀. Combining these results we see that if S is a locally compact Hausdorff space and Z is complete, then M_s(S, Z) is completely metrizable if and only if S is hemicompact.
Theorem 4 :  If S is a locally compact Hausdorff space and Z is complete, then M_s(S, Z) is a complete topological vector space.
Proof.  Consider  (fγ )γ ∈Γ  be a Cauchy net in M_s(S, Z). We state that, for every s ∈ S, (fγ (s))_{γ ∈ Γ} is a Cauchy net in K(Z) in terms  to the Hausdorff metric. Fix ϵ > 0 and s₀ ∈ S. Since (fγ )_{γ ∈Γ} is a Cauchy net, there exists  so that
f_γ0 – f_γ1 ∈ W(0, {s₀}, ϵ) whenever γ₀, γ₁ ≥ . Fix z₀ ∈ f_γ0(s₀). Let

for every s ∈ S. Since for all s ∈ S and f_γ0 is musco, it follows  that there exists a net (s_λ)_λ∈Λ that converges to s₀ in S, and a net (z_λ)_λ∈Λ converging to z₀ in Z so that  for every λ ∈ Λ. Accordingly for  there is, for each λ ∈ Λ, some w_λ ∈ f_γ1(s_λ) so that z_λ–w_λ ∈ f_γ0 –f_γ1(s_λ). There is a subnet of (w_λ)_λ∈Λ that converges to some w₀ ∈ f_γ1(s₀).  z₀–w₀ ∈ f_γ0 –f_γ1(s₀). Therefore d_Z(z₀, w₀) =d_Z(z₀ – w₀, 0) < ϵ. In the same manner  it follows that for each w₀ ∈ f_γ1(s₀) there is z₀ ∈ f_γ0(x₀) so that d_Z(z₀, w₀) < ϵ.Therefore H(f_γ0(s₀), f_γ1(s₀)) < ϵ whenever γ₀, γ₁ ≥ , which verifies our statement.

Since Z is complete, it follows that we may assume that the metric on Y is complete. Therefore K(Z) is complete with respect to the Hausdorff metric. Hence (f_γ (s))_{γ ∈Γ} converges to some K_s ∈ K(Z), with respect to the Hausdorff metric, for every s ∈ S. It is clear from the preceding argument that the convergence of (f_γ (s)_{)γ ∈Γ} to K_s is uniform on compact subsets of S. We state that the map g : S ∋ s → K_s ∈ K(Z) is quasi-minimal usco. Fix s₀ ∈ S and a compact neighbourhood V₀ of s₀. Consider U be an open set containing g(s₀). Since g(s₀) = K_s0 is compact, it follows from the Lebesgue Number Lemma  that there exists ϵ > 0 so that g(s₀) ⊆ U_ϵ(g(s₀₎) ⊆ U where U_ϵ(K) =  for any K ∈ K(Z). Since (f_γ (s))_{γ ∈Γ} converges uniformly on V₀ to g(s) in terms with the Hausdorff metric, there is γ_ϵ ∈ Γ so that f_γ (s) ⊆ U _ϵ/2 (g(s)) and g(s) ⊆ U _ϵ/2 (f_γ (s)) whenever s ∈ V₀ and γ ≥ γ_ϵ. Fix γ₀ ≥ γ_ϵ. Since f_γ0 is usco, there is V ∈V_s0 so that f_γ0(s) ⊆ U _ϵ/2 (g(s₀)) wherever s ∈ V. Without loss of generality, we may assume that V ⊆ V₀ such that U _{ϵ/ 2} (f_γ0(s)) ⊆ U_ϵ(g(s₀)) for every s ∈ V. Then g(s) ⊆ U_ϵ(g(s₀)) ⊆ U for all s ∈ V such that g is usco at s₀. Since s₀ ∈ S is arbitrary, it follows that g is usco on S. To see that g is quasi-minimal, consider some ϵ > 0, s₀ ∈ S and a compact neighbourhood V₀ of s₀.
Since (f_γ (s))_{γ ∈Γ} converges to g(s) uniformly on V₀ there is γ ∈ Γ such that f_γ (s) ⊂ U ϵ 3 (g(s)) and g(s) ⊂ U _{ϵ/ 3} (f_γ (s)) for all s ∈ V0. It follows that there is  s_V0 ∈ V₀ so that diam f_γ (s_V0) < ϵ/3 . Since g(s_V0) ⊂ U _{ϵ/ 3} (f_γ (s_V0)) it follows
that diam(g(s_V0)) < ϵ. Therefore the set D_ϵ = {s ∈ S : diam (g(s)) < ϵ} is dense in S. Hence D_ϵ contains an open and dense set. Indeed, if s ∈ D _ϵ/2 then there is V ∈ V_s such that V ⊆ D_ϵ. Therefore g is quasi-minimal. It remains to show that (f_γ )_{γ ∈Γ} converges to f = ⟨g⟩ in M_s(S, Z). Fix K ∈ K(S) and ϵ > 0. Consider K′ be a compact subset of S containing K in its interior. Recall that (f_γ (s))_{γ ∈Γ} converges uniformly to g(s) on K′, with respect to the Hausdorff metric.
Since f (s) ⊆ g(s) for every s ∈ S it follows that there is   such that f (s) ⊆ U _{ϵ/ 2} (f_γ (s)) for every s ∈ K′ and γ ≥ .
Therefore f ⊕(–f_γ )(s)∩B(0, (ϵ/2) ) ≠ ∅ for every s ∈ K′ and γ ≥ . There is the map hγ : C ⇒ Z defined as

is usco. Since h_γ ⊆ f ⊕ (–f_γ ) it follows  that h_γ ⊇ f – f_γ . Therefore f – f_γ (s) ⊆ B(0, ϵ) for all s ∈ K and . Since this holds for all K ∈ K(S) and ϵ > 0 it follows that (f_γ )_{γ ∈Γ} converges to f in M_s(S, Z).

Theorem 5 : M_s(S, Z) is metrizable if and only if S is hemicompact.
Proof. Assume that S is hemicompact. Then there exists a countable set K₀ = {K_n : n ∈ N} ⊆ K(S) so that every K ∈ K(S) is contained in a member of K₀. It is easy to see that {W(0, K_n, (1/m)) : m, n ∈ N} is a basis for τ_s at 0 ∈ M_k(S, Z).
Therefore M_s(S, Z) is metrizable. Assume that M_s(S, Z) is metrizable. Since C(S, Z), with the usual point-wise operations, is a linear subspace of M(S, Z),
it follows that C(S, Z), as a topological subspace of M_s(S, Z), is a Hausdorff topological vector space. In fact, it is easy to see that the subspace topology on C(S, Z) is the compact open topology. It therefore follows that C(S, Z) is metrizable in the compact open topology so that S is hemicompact.

Corollary (a).  If S is a locally compact Hausdorff space and Z is complete, then Ms(S, Z) is a completely metrizable if and only if S is hemicompact.

Comparision Between M_s(S, Z) And M_k(S, Z):

Previously,  we compared  the topologies τ_s and τ_k. And showed  that τ_s is finer than τ_k, with equality holding when S is locally compact. This result is then used, in combination with the results obtained in the preceding  operations for M_s(S, Z), to obtain corresponding results for M_k(S, Z). We also explain how our results generalize the known results.

Theorem 6 : The following statements are true.
(i) τ_s is finer than τ_k.
(ii) If S is locally compact, then τ_s = τ_k.
Proof. (i) Consider f ∈ M(S, Z), K ∈ K(S), ϵ > 0 and g ∈ W(0, K, ϵ). We state

that f +g ∈ W(f , K, ϵ). To see that this is so, consider an arbitrary point s₀ ∈ K and y₀ ∈ f +g(s₀). There is z₀ ∈ f (s₀) and w₀ ∈ g(s₀) such that y₀ = z₀ +w₀.
Then, since g(s₀) ⊂ B(0, ϵ), it follows that d_Z(y₀, z₀) = d_Z(z₀ + w₀, z₀) = d_Z(w₀, 0) < ϵ. Now take z₀ ∈ f (s₀). Let

Since f ′(s) ≠ ∅ for every s ∈ S and f  is musco, there exists a net (x_λ)_λ∈Λ converging to x₀ in S, and a net (z_λ)_λ∈Λ converging to z₀ in Z so that z_λ ∈ f ′(x_λ) for each λ ∈ Λ. By the definition of  f ′ there exists, for each λ ∈ Λ, a point w_λ ∈ g(xλ) so that y_λ = z_λ+w_λ ∈ f +g(x_λ). Since g is usco, There exist w₀ ∈ g(x₀) and a subnet of (w_λ)_λ∈Λ that converges to w₀. There exists that y₀ = z₀+w₀ ∈ f +g(x₀). Since g(x₀) ⊂ B(0, ϵ), the translation invariance of the metric on Z implies that d_Z(y₀, z₀) = d_Z(w₀, 0) < ϵ. Therefore H(f (x₀), f + g(x₀)) < ϵ.
Since this is true for every x₀ ∈ K our state has been proven so that W(0, K, ϵ)+f ⊆ W(f , K, ϵ). Hence  τ_s is finer than τ_k.
(ii) Now considering that S is locally compact. Consider f ∈ M(S, Z), K ∈ K(S) and ϵ > 0. Let K′ be a compact subset of  S containing K in its interior. We state that W(f , K′, 2ϵ ) ⊆ W(0, K, ϵ) + f . It is sufficient to show that h – f ∈ W(0, K, ϵ)
wherever h ∈ W(f , K′, (ϵ/2) ). Since S is locally compact, and therefore a Baire space, it follows that the set  D = {s ∈ S : diam (f (s)), diam (h(s)) = 0} is dense in S. The above definition  of addition in M(S, Z) implies that h – f (s) = h(s) – f (s) for every s ∈ D. Therefore d_Z(h(s) – f (s), 0) < (ϵ/2) for every s ∈ D ∩ K′. It therefore that  for every s ∈ Int(K′). Hence  the map g :  S ⇒ Z defined by

is usco. Since h⊕(–f ) is quasi-minimal usco and g, h–f ⊆ h⊕(–f ), it is that h–f ⊆ g. Hence h – f (s) ⊂ B(0, ϵ) such that H(h – f (s), {0}) < ϵ for every s ∈ K. Thus h – f ∈ W(0, K, ϵ).

Corollary (b). If S is locally compact, then the following statements are true.
(i) M_k(S, Z) is a Hausdorff topological vector space.
(ii) The topology on M_k(S, Z) is independent of the choice of compatible metric on Z.
(iii) If Z is locally convex, then M_k(S, Z) is locally convex.
(iv) M_k(S, Z) is metrizable if and only if S is hemicompact.
(v) If S is Hausdorff and Z is complete, then M_k(S, Z) is complete.
(vi) If S is Hausdorff and Z is complete, then M_k(S, Z) is completely metrizable if and only if S is hemicompact.

Holá explained that M_k(S, Y) is completely metrizable wherever Y a complete metric space and S a locally compact hemicompact space. Corollary (b) (v) gives a partial converse to this result. Now considering that the S is a Baire space, such that M(S, R) = D^∗(S). This being the case, Corollary (b) (i) and (iii) generalize the above operations, while Corollary (b) (iv)–(vi) also partially  generalizes it.

After satisfying the Generalization of the Concepts in terms os the Topology of Linear Spaces we proceed towards doing so for Quasilinear Spaces, as below:

Topological Quasilinear Spaces: Definitions and Properties

Allow us  to start this  by giving some concepts  and basic  results. For some topological space S, the  N_s stands denoting  the family of all neighborhoods of an s ∈ S. Consider  S be a topological vector space TVS, for short, s ∈ S and  G ⊂ S. Then G ∈ N_s if and only if G – s ∈ N₀ and s – G ∈ N₀. This is what called the localization principle of TVSs.
A set S is called a quasilinear space QLS, for short,  if a partial ordering relation  “≤”, an algebraic sum operation, and an operation of multiplication by real numbers are defined in it in such way that the following conditions hold for any elements s, y, z, u ∈ S, and any real scalars α, β:
s ≤ s,
s ≤ z          if s ≤ y, y ≤ z,
s=y           if s ≤ y, y ≤ s,
s + y= y + s,
s + (y + z) =   (s +  y) + z

there is an element 0 ∈ S such that s  0  s,

α · (β · s) =  (α · β) · s,
α · (s +  y) =  α · s + α · y

1 · s  = s,
0 · s = 0,
(α +  β) · s ≤ α · s +  β · s,
s + z ≤ y + v             if s ≤ y, z ≤ v,
α · s ≤ α · y      if s ≤ y.

A linear space is a QLS with the partial ordering relation “s ≤ y if and only if s  = y”. Perhaps the most popular example of nonlinear QLSs is the set of all closed intervals of real numbers with the inclusion relation “⊆” , algebraic sum operation
A+  B = {a  b : a ∈ A, b ∈ B}                                                                    (13)

and the real-scalar multiplication
λA = {λa : a ∈ A}                                                                                       (14)
Denoting this set by KCR. While another one is KR, the set of all compact subsets of real numbers. In general, KE and KCE stand for the space of all nonempty closed bounded and nonempty convex and closed bounded subsets of any normed linear space E, respectively. Both are QLSs with the inclusion relation and with a slight modification of addition as follows:
(15)

and with  real-scalar multiplication above.
Hence, K_C(E) = {A ∈ KE : A convex}.
Lemma 1  In a QLS S the element 0 is minimal, that is, s = 0 if s ≤ 0.
Definition . An element s ∈ S is called an inverse of an s ∈ S if s + s^’=  0. If an inverse element exists, then it is unique. An element s having an inverse is called regular; otherwise, it is called singular.
We prove later that the minimality is not only a property of 0 but also is shared by the other regular elements.
Lemma 2.  Suppose that each element s in the QLS S has an inverse element s^’ ∈ S. Then the partial ordering in S is determined by equality, the distributivity conditions hold, and, consequently, S is a linear space.
Corollary (c) :  In a real linear space, equality is the only way to define a partial ordering such that operation (13)  hold.
It will be assumed in what follows that –s  = -1(s). An element s in a QLS is regular if and only if s – s =  0 if and only if =  –s.
Definition . Suppose that S is a QLS and Y ⊆ S. Y is called a subspace of S wherever Y is a quasilinear space with the same partial ordering and the same operations on S.
Theorem 7 : Y is a subspace of a QLS S if and only if for every s, y ∈ Y and α, β ∈ R, αs + βy ∈ Y.
Proof of this theorem is quite similar to its classical linear algebraic counterpart.
Consider  S  be a QLS and Y be a subspace of S. Suppose that each element s in Y has an inverse element  ∈ Y; then the partial ordering on Y is determined by the equality. In this case the distributivity conditions hold on Y, and Y is a linear subspace of S.
Definition. Consider S be a QLS. An element s ∈ S is said to be symmetric provided that –s =  s, and X_b denotes the set of all such elements. Further, X_r and X_s stand for the sets of all regular and singular elements in S, respectively.

Theorem 8 :  X_r, X_d, and X_s ∪ {0} are subspaces of S.

Proof.  X_r is a subspace since the element  is the inverse of λs + y.
X_s ∪ {0} is a subspace of S. Let s, y ∈ X_s ∪ {0} and λ ∈ R. The assertion is clear for s= y= 0. Let s ≠  0 and suppose that  s+ λy ∉ Xs ∪ {0}, that is, (s+ λy)+ u = 0 for some u ∈ S. Then  s+( λy+ u)= 0 and so  = λy + u. This implies that s ∈ X_r. Analogously we obtain y ∈ X_r if y ≠  0. This contradiction shows that s + λy ∈ X_s ∪ {0}.
The proof for X_d is similar.
X_r, X_d, and X_s ∪ {0} are called regular, symmetric, and singular subspaces of S, respectively.

Example. Consider S =K_CR and Z= {0} ∪ {a, b : a, b ∈ R and a≠ b}. Z is the singular subspace of S. However, the set {{a} : a ∈ R} of all singletons constitutes X_r and is a linear subspace of S. Factually, for any normed linear space E, each singleton {a}, a ∈ E is identified with a, and hence E is considered as the regular subspace of both KCE and KE.

Lemma 3 The operations of algebraic operations of addition and scalar multiplication are continuous with respect to the Hausdorff metric. The norm is continuous function with respect to the Hausdorff metric.

Lemma 4  (a) Suppose that s_n → s₀ and y_n → y₀, and that s_n ≤ y_n for any positive integer n. Then s₀ ≤ y₀.

 (b) Suppose that s_n → s₀ and z_n → s₀. If s_n ≤ y_n ≤ z_n for any n, then y_n → s₀.

(c) Suppose that s_n + y_n → s₀ and y_n → 0; then s_n → s₀.

Example Consider S  be a real complete normed linear space a real Banach space. Then S is a complete normed quasilinear space with partial ordering given by equality. Conversely, if S is complete normed quasilinear space and any s ∈ S has an inverse element  ∈ S, then S is a real Banach space, and the partial ordering on S is the equality. In this case  , it is notable that  for nonlinear QLS, in general.
For example, if E is a Banach space, then a norm on K(E)  is defined by  . Then KE and K_CE are normed quasilinear spaces. In this case the Hausdorff metric is defined as usual:
,                                    (15)

Defining the Topology of Quasilinear Spaces:

Definition. A topological quasilinear space TQLS, for short S is a topological space and a quasilinear space such that the algebraic operation of addition and scalar multiplication are continuous, and, following conditions are satisfied for any s, y ∈ S:
for any U ∈ N0, s ≤ y and y ∈ U implies s ∈ U,                                         (16)
for any U ∈ N_s, y ∈ U ⇐⇒ there IS some V ∈ N0 satisfying  s+V ⊆ U,
such that s ≤ y+a for some a ∈ V or y ≤ s+b for some b ∈ V.                   (17)
Any topology τ, which makes S, τ be a topological quasilinear space, will be called a quasilinear topology. The ABOVE conditions  provide necessary harmony of the topology with the ordering structure on S.

Example   Consider S be a TVS. Then, for any s, y ∈ S and for any U ∈ N_s, y ∈ U if and only if there exists a neighborhood V of 0 satisfying  s+ V ⊆ U such that s=  y + a for some a ∈ V or  y= s+ b for some b ∈ V . In fact, this is true by the localization principle of TVSs since U – s and  s – U are neighborhood of 0. Hence we  obtain desired V by taking V=  U – s or V=  s – U. This provides the previous conditions.  Hence, S is a TQLS.
We later show that some TQLS may not satisfy the localization principle.

Remark. In condition 3.2, for some U ∈ N_s, we may find a V ∈ N₀ satisfying   s+V ⊆ U such that both s ≤ y+a and y ≤ s + b for some a, b ∈ V . This comfortable situation depends on the selection of U. However, we may not find such a suitable V ∈ N₀ for some U ∈ N_x even in TVS.

Esample. Consider real numbers with usual metric. Take s =3, y= 5, and U = [2,7] ∈ N_s. Then any V ∈ N₀ satisfying 3+V ⊆ U must be a subset of  [ -1, 4]. Further 3 =5+a and 5=  3+b gives a =  -2, b= 2, and hence V can only include b.

Remark . In a semimetrizable TQLS the condition can be reformulated by balls as follows:
for any ε > 0, s ≤ y and y ∈ Sε0 implies s ∈ S_ε(0)                                             (18)
equivalently,

s ≤ y implies d(s, 0) ≤ d(y, 0),
for any ε > 0, y ∈ S_ε s⇐⇒ there exists some Sε0,
with s  Sε0 ⊆ Sεs such that s ≤ y  a for some a ∈ S_ε0,
                           or y ≤ s+b for some b ∈ S_ε0.                                 (19)

A TQLS with a semimetrizable quasilinear topology will be called a (semi)metric QLS.

Theorem 9 : Let S be a TQLS. Then X_r and X_d are closed in S.
Proof. {s_i} is a net in S_r converging to an s ∈ S. By the continuity of algebraic operations  –s_i → –s and s_i –s_i → s –s. This means s –s= 0 since s_i –s_i =  0 for each i, whence s ∈ S_r.
The proof is easier for X_d.
The result of this theorem may not be true for X_s ∪ {0}. Consider S =  K_C(R) and define  for each n ∈ N.

Then s_n → {1} ∉ X_s ∪ {0}.

Definition Consider S be a quasilinear space. A paranorm on S is a function     p : S → R satisfying the following conditions. For every s, y ∈ S,
(i) p(0) = 0

(ii) p(s) ⩾ 0

(iii) p(-s) = p(s)

(iv) p(s+y) ⩽ p(s) + p(y)

(v) if {t_n} is a sequence of scalars with t_n → t and {s_n} ⊂ S with p(s_n) → p(s), then p(t_ns_n) → p(ts) and (continuity of scalar multiplication)
(vi) if s ≤ y, then p(s) ≤ p(y).
The pair (S, p) with the function p satisfying the conditions (i) to (vi) is called a paranormed QLS. There exist that if any s ∈ S has an inverse element  ∈ S, then the concept of paranormed quasilinear space coincides with the concept of a real paranormed linear space. This paranorm is called total if, in addition, we have p(s) = 0 ⇐⇒ s  0,

if for any ε > 0 there exists an element s_ε ∈ S such that, s ≤ y+ s_ε and p(s_ε) ≤ ε, then s ≤ y. 

                                                                                                                       (20)

The equality
(21)
defines a semimetric on a paranormed quasilinear space S. d is metric wherever p is total.

Now, consider  p be total and d(s, y) = 0. Then for any ε > 0 there exist elements,  ∈ S such that s ≤ y+ , and y ≤ s+  , for   ≤ ε, i  1, 2. Hence the totality conditions imply that s ≤ y and y ≤ s, that is, s  y.

Further, we have the inequality d(s+, y) ≤ p(s – y).
Note that these definitions are inspired from the definitions about normed
quasilinear spaces. The proofs of some facts given here are quite similar to that of Aseev’s corresponding results.
If the first condition in the definition of norm in a QLS is relaxed into the condition
‖s‖ ≥ 0         if s≠  0                                                                                   (22)
and if the norm is removed, we obtain the definition of a seminorm.
A quasilinear space having  seminorm is called a seminormed QLS. Similarly in linear spaces it can proved that a seminorm on a QLS is a paranorm. Thus we have the following implication chain among the kinds of QLS :
normed-seminormed QLS ⇒ total paranormed-paranormed  QLS ⇒ metric-semimetric QLS ⇒ Hausdorff TQLS.

Definition. Consider ( S, d)  be a semimetric QLS and s be an element of S. Then, the nonnegative number
ρ(s) =  d(s – s, 0)                                                                                      (23)
is called diameter of s. For every regular element s, ρ(s)= 0 since s – s =  0. Hence this definition is redundant in linear spaces. In addition it should not be confused with the classical notion of the diameter of a subset in a semimetric space for which it is defined by δ(U) = sup_s,y∈Ud(s, y)  for any U ⊂ S.
For example, in K_C(R),[ -1, 3] ∈ KCR and  ρ([-1, 3]) = h([-1, 3] –[ -1, 3], 0)
=  h
= sup |a| =  4

                     a ∈ [-4,4].                                                                             (24)
However, for the singleton subset U = {[-1, 3]} of K_C(R), δ(U)= 0.
result is half of the localization principle of TVS.

Theorem 10:  Consider S be a TQLS, s ∈ S, and U is a set containing 0. If s + U ∈ N_s,,  then U ∈ N₀.
Proof. The proof is just an application of the fact that the translation operator fs : S → S, f_s(v) =  v + s, is continuous by the continuity of the algebraic sum operation.  Though the converse of this theorem is true in almost all the TVSs, it may not be true in some TQLSs.

Example Consider K_C(R) again and let it be closed unit ball S₁(0). Now, for      s = [2, 3] ∈ K_C(R), we show that s+ S₁(0) is not the neighborhood of s. A careful observation shows that s+ S₁(0)  do not contain elements intervals for which the diameter is smaller than 1. However, every s-centered ball S_r(s)  with radius r contains a singleton if r ≥ ρ(s)/2 = ½ and contains an interval such as    [2+( r/2), 3 –( r/2)] if r < 1/2 since h([2, 3] ,[ 2+( r/2), 3 – (r/2)]=(r.2)<r     (25)
That shows that, S_r(s) contains elements with diameter smaller than 1. However, neither a singleton nor such an element belongs to s+ S_r(0). This implies,       S_r(s)  s+S_r(0) for every r > 0. Eventually, the set s+S₁(0) cannot contain an s-centered ball. Thus, the localization principle may not be satisfied about a singular element in K_C(R). The example points that translation by a singular element destroys the property of being a neighborhood in a TQLS.

Theorem 11: Consider S be a TQLS and s ∈ X_r. Then U ∈ N₀ ⇔ s+ U ∈ N_s.
Proof. Consider again the operator f_s in the Theorem 10. If this be the case the inverse  exists and is the continuous operator f_–s. Hence f_s is a homeomorphism hence preserves the neighborhoods.

CONCLUSIONS

This paper has significantly advanced the understanding of topological properties within the framework of quasilinear spaces and their crucial applications to set-valued maps. Our work successfully generalizes Aseev’s foundational concept of normed quasilinear spaces to the more encompassing domain of topological quasilinear spaces, thereby broadening the theoretical landscape for such structures.

A central contribution is the rigorous demonstration that the space of minimal semi-continuous compact-valued maps, M(S,Z), can be endowed with a linear space structure, even when S is an arbitrary topological space and Z a metrizable topological vector space. This finding is particularly impactful as it extends previous results that were confined to more restrictive conditions on S, such as being a Baire space, necessitating the development of novel techniques based on the metric characterization of quasi-minimal upper semi-continuous compact-valued maps.

Furthermore, the paper provides a detailed topological analysis of M(S,Z) when equipped with the topology of strong uniform convergence on compact subsets of S, denoted as M_s(S,Z). We establish several fundamental properties of this space, including its completeness under the conditions that S is a locally compact Hausdorff space and Z is complete. A key result is the characterization of its metrizability, proving that M_s(S,Z) is metrizable if and only if S is hemicompact. This comprehensive characterization provides essential tools for further research into the analytical properties of such function spaces.

The comparative study of the strong uniform topology (τ_s) and the topology of uniform convergence on compact sets (τ_k) reveals that τ_s is consistently finer than τ_k, with these topologies coinciding when S is locally compact. This elucidates the precise relationship between different modes of convergence in this generalized setting. Moreover, our investigation into the principle of localization highlights that its universal validity, typical in topological vector spaces, may not extend to all topologically quasilinear spaces, especially concerning singular elements. This nuance is critical for avoiding pitfalls in future theoretical developments.

Collectively, these findings offer fresh and profound insights into the intricate topology of quasilinear spaces, significantly enriching the fields of multivalued mappings and set-valued analysis. The established linear and topological structures provide a robust foundation for advancing functional analysis in non-linear contexts, paving the way for new applications in areas dealing with uncertainty and set-valued phenomena.

REFERENCES

1. L’. Holá, Complete metrisability of generalized compact-open topology, Topology Appl. 91 (1999) 159–167.
2. L’. Holá, Spaces of dense continuous forms, usco and minimal usco maps, Set-Valued Anal. 11 (2003) 133–151.
3. L’. Holá, Spaces of densely continuous forms, Topology Proc. 31 (2007) 145–150.
4. Anguelov, J.H. van der Walt, Algebraic and topological structure of some spaces of set-valued maps, Comput. Math. Appl. 66 (2013) 1643–1654.
5. M. Fort, Points of continuity of semicontinuous functions, Publ. Math. Debrecen 2 (1951) 100–102.
6. M. Borwein, S. Fitzpatrick, P. Kenderov, Minimal convex uscos and monotone operators on small sets, Canad. J. Math. 43 (1991) 461–477.
7. Hu, N. Papageorgiou, Handbook of Multi-Valued Analysis, Vol. I: Theory, Kluwer, Dordrecht, 1997.
8. M. Borwein, W.B. Moors, Essentially smooth Lipschitz functions, J. Funct. Anal. 149 (1997) 305–351.
9. R. Munkres, Topology, second ed., Prentice Hall, New Jersey, 2000.
10. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952) 484–487.
11. B. Gauld, Z. Piotrowski, On Volterra spaces, Far East J. Math. Sci. 1 (1993) 209–214.
12. Tréves, Topological Vector Spaces, Distributions and Kernels. Unabridged Republication of the 1967 Original, Dover Publications, Mineola, New York, 2006.
13. B. Price, On the completeness of a certain metric space with an application to Blaschke’s selection theorem, Bull. Amer. Math. Soc. 46 (1940) 278–280.
14. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480–495.
15. M. Aseev, “Quasilinear operators and their application in the theory of multivalued mappings,”Proceedings of the Steklov Institute of Mathematics, no. 2, pp. 23–52, 1986.
16. A. Rojas-Medar, M. D. Jimenez-Gamero, Y. Chalco-Cano, and A. J. Viera-Brand ´ ao, “Fuzzy ˜quasilinear spaces and applications,” Fuzzy Sets and Systems, vol. 152, no. 2, pp. 173–190, 2005.
17. Ratschek and J. Rokne, New Computer Methods for Global Optimization, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, Chichester, UK, John Wiley & Sons, New York, NY,
    USA, 1988.
18. Markov, “On the algebraic properties of convex bodies and some applications,” Journal of Convex Analysis, vol. 7, no. 1, pp. 129–166, 2000.
19. Markow, “On quasilinear spaces of convex bodies and intervals,” Journal of Computational and Applied Mathematics, vol. 162, no. 1, pp. 93–112, 2004.
20. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, NY, USA, 1978.
21. Markov, Biomathematics and interval analysis: a prosperous marriage, in: M.D. Todorov, Ch.I. Christov (Eds.), AIP Conference Proceedings, in: Amitans’2010, vol. 1301, American Institute of Physics, Melville, New York, 2010, pp. 26–36.
22. M. Aseev, Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math. 167 (1986) 23–52.
23. Anguelov, O.F.K. Kalenda, The convergence space of minimal usco mappings, Czechoslovak Math. J. 59 (2009) 101–128.
24. T. Hammer, R.A. McCoy, Spaces of densely continuous forms, Set-Valued Anal. 5 (1997) 247–266.
25. A. McCoy, Fell topology and uniform topology on compacta on spaces of multifunctions, J. Rostocker Math. Kolloq. 51 (1997) 127–136.
26. A. McCoy, Spaces of semi-continuous forms, Topol. Proc. 23 (1998) 249–275.
27. Talo and F. Bas ¨ ¸ar, “Quasilinearity of the classical sets of sequences of fuzzy numbers and some related results,” Taiwanese Journal of Mathematics, vol. 14, no. 5, pp. 1799–1819, 2010.
28. Lakshmikantham, T. Gana Bhaskar, and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, 2006.
29. P. Aubin, H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, Basel, Berlin, 1990.
30. P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley, 1984.
31. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, 1983.
32. Klein, A.C. Thompson, Theory of Correspondences, Including Applications to Mathematical Economics, Wiley, New York, 1984.