Initial and the final T -topogenous structures

Abstract

This paper is devoted to study the initial and the final T-topogenous structures of the T-topogenous spaces, defined by the author in 2012. In this manuscript, we show that all initial (final) lifts in the category of T-topogenous spaces and the initial (final) T-topogenous structures exist. We introduce characterizations of the initial T-topogenous structures as special initial T-topogenous structures. The subspaces and product spaces of these T-topogenous spaces can be also characterized. Moreover, we show that the I-topology associated to the initial T-topogenous structure of a family of T-topogenous spaces coincides with the initial I-topology of the family of I-topologies associated to these T-topogenous spaces.

1 Introduction

The notions of T-topogenous spaces was introduced in [7, 15]. In this manuscript, we introduce for a family of T-topogenous spaces ((Y_r, {η_r})) _r∈Λ and a family (f_r) _r∈Λ of mappings f_r of a set X into Y_r, the coarsest T-topogenous structure ζ on X such that all mappings f_r: (X, {ζ}) ⟶ (Y_r, {η_r}) are syntopogenously continuous. Also, we show that the coarsest T-topogenous structure {ζ} provides the initial lifts in the category of T-topogenous spaces and their syntopogenously continuous functions, hence it is the initial T-topogenous structure in the categorical sense. That is, all initial lifts and all initial T-topogenous structures exist in the category of T-topogenous spaces and therefore the category of T-topogenous spaces is topological [1]. The subspaces and the product spaces of T-topogenous spaces in the categorical sense are special initial T-topogenous spaces, hence these spaces exist and can be characterized. Since the concrete category of T-topogenous spaces is topological, then all final lifts and all final T-topogenous structures also exist. The quotient spaces and the sum spaces ofT-topogenous spaces can be characterized as special final T-topogenous spaces.

In [12], Lowen introduced the notion of initial I-topology. In this paper we illustrate the close relationship between the initial I-topology and the initial T-topogenous structure, since an I-topology τ ({ η }) is associated with each T-topogenous structure {η}, we show that the I-topology associated with the initial T-topogenous structure of a family ({η_r} _{)_r∈Λ} of T-topogenous structures {η_r} coincides with the initial I-topology of the family (τ {η_r} _{)_r∈Λ} of I-topologies τ ({η}) associated with {_{η
_r}}. Also, we define a functor from the category of T-topogenous spaces, to the category of I-topological spaces which preserves initial lifts.

Motivation of the manuscript: The notions of initial lift and final lift play an important role in the category theory. Here, we introduce these notions of the category of T-topogenous spaces and their syntopogenously continuous functions.

1.1 Preliminaries

In this section, we give some definitions of fuzzy sets, I-topological spaces and T-topogenous spaces presented in [7 , 19] which will be needed in the following sections.

A triangular norm (cf. [17]) is a binary operation on the unit interval I = [0,1] that is associative, symmetric, isotone in each argument and has the neutral element 1. The triangular conorm of a triangular norm T is the binary operation T^* on the unit interval I given by: $α T^{*} β = 1 - [(1 - α) T (1 - β)], α, β \in I .$

A fuzzy set in a universe set X, introduced by Zadeh in [19], is a function λ : X → I. The collection of all fuzzy sets of X is denoted by I^X. The height of a fuzzy set λ is the following real number: $hgt λ = \sup {λ (x) : x \in X} .$

We shall often need to consider a subset H ⊆ X as a fuzzy subset of X, said to be a crisp fuzzy subset of X, which we shall denote by the symbol 1_H. We do this by identifying 1_H with its characteristic function. We also denote the constant fuzzy set of X with value α ∈ Iby $\underline{\underline{α}}$ .

For an I-topology τ on a set X (follow Lowen’s definition [11]) we mean a subset of I^X which contains all constant fuzzy sets $\underline{\underline{α}}$ where α ∈ I and is closed with respect to the finite infima and arbitrary suprema. The pair (X, τ) is called an I-topological space.

Lemma 1.1. [5] Let f: X → Y be a function. Then for all μ ∈ I^x, $hgt f (μ) = hgt μ .$

Lemma 1.2. [6] Let f: X → Y be a function. Then for all μ, λ ∈ I^x, we have $hgt [f (μ) T f (λ)] \geq hgt (μ T λ) .$

Here we will give an ideas of T-topogenous structures.

Definition 1.1. [7] A T-topogenous order on a set X is a function ζ: I^X × I^X → I, which satisfies, for every μ, λ, ν ∈ I^X and α ∈ I, the following:

$ζ (\underline{\underline{1}}, \underline{\underline{α}}) = α$ and $ζ (\underline{\underline{α}}, \underline{\underline{0}}) = 1 - α$ ;

ζ (μ ∨ λ, ν) = ζ (μ, ν) ∧ ζ (λ, ν) and ζ (μ, λ ∧ ν) = ζ (μ, λ) ∧ ζ (μ, ν);

If ζ (μ, λ) >1 - (θ T β) for some θ, β ∈ I₀ =]0, 1], there is C ⊆ X such that ζ (μ, 1_C)≥ 1 - θ and ζ (1_C, λ) ≥1 - β;

$ζ (μ, λ) \leq 1 - hgt [μ T (\underline{\underline{1}} - λ)]$ ;

$ζ (\underline{\underline{α}} T μ, λ) = (1 - α) T^{*} ζ (μ, λ)$ $= ζ (μ, \underline{\underline{(1 - α}}) T^{*} λ) .$

Definition 1.3. [7] A T-syntopogenous structure on a set X is a family ℘ of T-topogenous orders on X satisfying the following conditions:

℘ is directed in the sense that, given ζ, η∈ ℘ there is ξ∈ ℘ such that ξ ≥ ζ ∨ η;

for every ζ∈ ℘ and ɛ ∈ I₀_, there is ζ_ɛ∈ ℘ such that (ζ_ɛo_T ζ_ɛ) + $\underline{\underline{ɛ}} \geq ζ$ .

Where the operation o_T (see [7]) defined by (ξ o_T η)(μ, λ) = $sup_{C \subseteq X}$ [η(μ, 1_C) T ξ (1_C, λ)].

The pair (X, ℘) is said to be a T-syntopogenous space. If a T-syntopogenous structure ℘ on X consists of a single T-topogenous order, then ℘ is called a T-topogenous structure and (X, ℘) a T-topogenous space [15].

The I-topology τ ({ζ}) on X, induced by {ζ} is given by the fuzzy interior operator [7, 15]: μ° (X) = ζ (1_x, μ), μ ∈ I^x, x ∈ X.

The concepts of a syntopogenous continuous map has main role in the theory of T-topogenous spaces and is defined as follows: Given T-topogenous spaces (X, {ζ}), (Y, {η}) and a function f : X → Y, we say that f is syntopogenously continuous with respect to {ζ} and {η} (cf. [7]), if it satisfies $η (ν, ρ) \leq ζ (f^{\leftarrow} (ν), f^{\leftarrow} (ρ)), ν, ρ \in I^{Y},$ (1) $\begin{matrix} Equivalently, η (f (μ), \underset{=}{1} - f (λ)) \leq \\ ζ (μ, \underset{=}{1} - λ), μ, λ \in I^{X} . \end{matrix}$ (2)

Given two T-topogenous orders ζ₁, ζ₂ on a set X, we say that ζ₁ is coarser than ζ₂ (and ζ₂ is finer than ζ₁), if $ζ_{1} (μ, λ) \leq ζ_{2} (μ, λ), \forall μ, λ \in I^{X} .$ (3)

Proposition 1.1. [7] If ζ₁ and ζ₂ are T -topogenous orders defined on a set X, then ζ₁ is coarser than ζ₂ implies τ ({ζ₁}) ⊆ τ ({ζ₂}).

1.2 Initial and final T-topogenous structures

In this section, we introduce the concepts of the initial T-topogenous structure of a family of T-topogenous spaces. Also, we characterize the initial structures of the notion of T-topogenous spaces presented in [15].

At first, consider the case of one mapping.

Theorem 2.1. Assume that f : X → Y is a mapping and η a T -topogenous order on Y. Define the mapping $\begin{matrix} f^{\leftarrow} (η) : I^{X} \times I^{X} \to I, by : \\ (f^{\leftarrow} (η)) (μ, λ) = η (f (μ), \underset{=}{1} - f (\underset{=}{1} - λ)), \\ μ, λ \in I^{X} . \end{matrix}$

Then f^← (η) is the coarsest T-topogenous order on X such that the mapping f is syntopogenouslycontinuous.

Proof. Obviously, f^← (η) is a T-topogenous order on X [7, Proposition 3.15]. Moreover, the syntopogenous continuity of the function f : (X, {f^← (η)}) → (Y, {η}) follows immediately from (2).

Now, let ζ be a T-topogenous order on X and μ, λ ∈ I^x.

Then by syntopogenously continuous of f : (X, {ζ}) → (Y, {η}), we get $\begin{matrix} (f^{\leftarrow} (η)) (μ, λ) & = & η (f (μ), \underset{=}{1} - f (\underset{=}{1} - λ)) \\ \leq & ζ (μ, λ), \end{matrix}$ which shows, by (3) that, f^← (η) is coarser than ζ.□

Assume now a family ((Y_r {η_r})) _r∈Λ of T-topogenous spaces and a family of mappings (f_r : X → (Y_r, {η_r})) _r∈Λ are given, where Λ may be any class.

Theorem 2.2. Let ((Y_r, {η_r})) _r∈Λ be a family of T -topogenous spaces, X be a set and for each r ∈ Λ, there is a function f_r : X → (Y_r, {η_r}). Define ζ_o: I^x × I^x → I by: for every μ, λ ∈ I^x, $\begin{matrix} ζ_{o} (μ, λ) = max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} \\ - f_{r} (\underset{=}{1} - λ_{j}))} . \end{matrix}$ where the maximum is taken over all finite families (μ_i), (λ_j) of fuzzy sets on X such that $μ = \underset{i}{\lor} μ_{i}$ , $λ = \underset{j}{\land} λ_{j}$ . Then

{ζ_o} is the coarsest T -topogenous structure on X such that each f_r is syntopogenouslycontinuous.

A mapping g from a T -topogenous space (Z, {ζ_o}) to (X, {ζ}) is syntopogenously continuous if and only if each f_ro g: (Z, {ζ}) → (Y_r, {η_r}) is syntopogenously continuous.

Proof.

First, we show that ζ_o is a T-topogenous order on X. Let μ, ν, λ ∈ I^x and α ∈ I, so there are finite families (μ_i), (ν_k), (λ_j) of fuzzy sets such that $μ = \underset{i}{\lor} μ_{i}$ , $ν = \underset{k}{\lor} ν_{k}$ , $λ = \underset{j}{\land} λ_{j}$ . Then we have the following:

Obviously holds. $\begin{matrix} (TT 2) ζ_{o} (μ \lor ν, λ) \\ = max {inf_{i, k, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i} \lor ν_{k}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = max {inf_{i, k, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}) \\ \lor f_{r} (ν_{k}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = max {inf_{i, k, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j})) \\ \land η_{r} (f_{r} (ν_{k}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))]} \\ \leq max {inf_{i, k, j} \underset{r \in Λ}{[max} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j})) \\ \land max_{r \in Λ} η_{r} (f_{r} (ν_{k}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))]} \\ = max {\underset{i, j}{[inf} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))] \\ \land \underset{k, j}{[inf} max_{r \in Λ} η_{r} (f_{r} (ν_{k}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))]} \\ \leq max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ \land max inf_{k, n} max_{r \in Λ} η_{r} (f_{r} (ν_{k}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = ζ_{o} (μ, λ) \land ζ_{o} (ν, λ) . \end{matrix}$

For the converse inequality, let ɛ ∈ I₀ be such that ɛ > ζ_o (μ ∨ ν, λ).

Then for every finite families (μ_i), (ν_k), (λ_j) offuzzy sets such that $μ = \underset{i}{\lor} μ_{i}$ , $ν = \underset{k}{\lor} ν_{k}$ , $λ = \underset{j}{\land} λ_{j}$ and some $\underline{i}$ , $\underline{j}$ , $\underline{k}$ and all r ∈ Λ, we have $\begin{matrix} ɛ > η_{r} (f_{r} (μ_{\underline{i}}) \lor f_{r} (ν_{\underline{k}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}})) \\ = η_{r} (f_{r} (μ_{\underline{i}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}})) \land \\ η_{r} (f_{r} (ν_{\underline{k}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}})), \end{matrix}$ that is $\begin{matrix} ɛ > η_{r} (f_{r} (μ_{\underline{i}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}})) or \\ ɛ > η_{r} (f_{r} (ν_{\underline{k}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}})), \end{matrix}$ hence $\begin{matrix} ɛ & \geq & max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}}))} \\ = & ζ_{o} (μ, λ), or \\ ɛ & \geq & max {inf_{k, j} max_{r \in Λ} η_{r} (f_{r} (ν_{\underline{k}}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{\underline{j}}))} \\ = & ζ_{o} (ν, λ), \end{matrix}$ therefore ɛ ≥ ζ_o (μ, λ) ∧ ζ_o (ν, λ).

Which implies ζ_o (μ ∨ ν, λ) ≥ ζ_o (μ, λ) ∧ ζ_o (ν, λ). This proves ζ_o (μ ∨ ν, λ) = ζ_o (μ, λ) ∧ ζ_o (ν, λ). In the same manner, we show ζ_o (μ, λ ∧ ν) = ζ_o (μ, λ) ∧ ζ_o (μ, ν).

(TT3) Let ζ_o (μ, λ) >1 - (θ T β) for some θ,β ∈ I₀.

Then we get $η_{t} (f_{t} (μ_{i}), \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) > 1 - (θ T β)$ , for some t ∈ Λ and all i, j.

For this index t ∈ Λ, we find C_ij ⊆ Y_tsuch that η_t (f_t (μ_i) , 1_{C
_ij}) ≥1 - θ and η_t (1_{C
_ij}, $\underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) \geq 1 - β$ , hence $inf_{i, j} η_{t} (f_{t} (μ_{i}), 1_{C_{ij}})$ ≥1 - θ and $inf_{i, j} η_{t} (1_{C_{ij}}, \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) \geq 1 - β$ .

By taking $1_{C} = \underset{j}{\land} \underset{i}{\lor} f_{t}^{\leftarrow} (1_{C_{ij}}) \in 2^{x}$ , we get $\begin{matrix} ζ_{o} (μ, 1_{C}) = max {inf_{i, j} max_{r \in Λ} \\ η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - \underset{i}{\lor} f_{t}^{\leftarrow} (1_{C_{ij}})))} \\ \geq inf_{i, j} η_{t} (f_{t} (μ_{i}), \underset{=}{1} - f_{t} (\underset{=}{1} - \underset{i}{\lor} f_{t}^{\leftarrow} (1_{c_{ij}}))) \\ = inf_{i, j} η_{t} (f_{t} (μ_{i}), \underset{=}{1} - f_{t} f_{t}^{\leftarrow} (\underset{=}{1} - \underset{i}{\lor} 1_{c_{ij}})) \\ \geq inf_{i, j} η_{t} (f_{t} (μ_{i}), \underset{i}{\lor} 1_{c_{ij}}), by (TT 2) \\ \geq inf_{i, j} η_{t} (f_{t} μ_{i}), 1_{C_{ij}}), by (TT 2) again \\ \geq 1 - θ, and \end{matrix}$ $\begin{matrix} ζ_{o} (1_{c}, λ) = max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (\underset{j}{\land} f_{t}^{\leftarrow} (1_{c_{ij}})), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ \geq inf_{i, j} η_{t} (f_{t} (\underset{j}{\land} f_{t}^{\leftarrow} (1_{c_{ij}}), \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) \\ = inf_{i, j} η_{t} (f_{t} (f_{t} f_{t}^{\leftarrow} (\underset{j}{\land} 1_{c_{ij}}), \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) \\ \geq inf_{i, j} η_{t} (\underset{j}{\land} 1_{c_{ij}}), \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})) by (TT 2) \\ \geq inf_{i, j} η_{t} (1_{C_{ij}}, \underset{=}{1} - f_{t} (\underset{=}{1} - λ_{j})), by (TT 2) \\ \geq 1 - β . \end{matrix}$ $\begin{matrix} (TT 4) ζ_{o} (μ, λ) = max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ \leq max {inf_{i, j} max_{r \in Λ} {1 - hgt [f_{r} (μ_{i}) T \\ f_{r} (\underset{=}{1} - λ_{j})]}} by (TT 4) \\ \leq max {inf_{i, j} [1 - hgt (μ_{i} T (\underset{=}{1} - λ_{j}))]}, \\ by Lemma 1.2 \\ = max {1 - [max_{i, j} hgt (μ_{i} T (\underset{=}{1} - λ_{j}))]} \\ = max {1 - hgt [(\underset{i}{\lor} μ_{i}) T \underset{j}{\lor} (\underset{=}{1} - λ_{j}))]} \\ = max {1 - hgt [(\underset{i}{\lor} μ_{i}) T (\underset{=}{1} - \underset{j}{\land} λ_{j}))]} \\ = 1 - hgt [μ T (\underset{=}{1} - λ)] . \end{matrix}$ $\begin{matrix} (TT 5) ζ_{o} (\underline{\underline{α}} T μ, λ) = max {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (\underset{=}{α} T μ_{i}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = max {inf_{i, j} max_{r \in Λ} η_{r} (\underset{=}{α} T f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = max {inf_{i, j} max_{r \in Λ} [(1 - α) T^{*} η_{r} (f_{r} (μ_{i}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))]} \\ = max {(1 - α) T^{*} [inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))]} \\ = (1 - α) T^{*} max \\ {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = (1 - α) T^{*} ζ_{o} (μ, λ) . \end{matrix}$

Analogously, we can show $ζ_{o} (μ, (\underline{\underline{1 - α}}) T^{*} λ) = (1 - α) T^{*} ζ_{o} (μ, λ)$ .

This proves ζ_o is a T-topogenous order on X.

Next, let ζ^∖ be a T-topogenous order on X and μ, λ ∈ I^x.

Then by syntopogenously continuous of f_r : X → (Y_r, {η_r}), we have for every r ∈ Λ : $ζ^{∖} (μ, λ) \geq η_{r} (f_{r} (μ), \underset{=}{1} - f_{r} (\underset{=}{1} - λ)), by (2)$ That is, $ζ^{∖} (μ, λ) \geq max_{r \in Λ} η_{r} (f_{r} (μ), \underset{=}{1} - f_{r} (\underset{=}{1} - λ)) .$ Hence $\begin{matrix} ζ^{∖} (μ, λ) & = & ζ^{∖} (\underset{i}{\lor} μ_{i}, \underset{j}{\land} λ_{j}) \\ = & \underset{i, j}{\land} ζ^{∖} (μ_{i}, λ_{j}), by (TT 2) twice \\ \geq & inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j})) . \end{matrix}$ Thus $\begin{matrix} ζ^{∖} (μ, λ) \geq max \\ {inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (μ_{i}), \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))} \\ = ζ_{o} (μ, λ) . \end{matrix}$ Therefore, by (3), we get ζ_o is coarser than ζ^∖.

(ii) Let (Z, {ζ}) be a T-topogenous space andg: (Z, {ζ}) → (X, {ζ_o}) be a function.

If g is syntopogenously continuous, then by (2), we have for all μ, λ ∈ I^z and all r ∈ Λ, $\begin{matrix} ζ (μ, \underset{=}{1} - λ) & \geq & ζ_{o} (g (μ), \underset{=}{1} - g (λ)) \\ \geq & η_{r} (f_{r} (g (μ)), \underset{=}{1} - f_{r} (g (λ))) \\ = & η_{r} ((f_{r} o g) (μ), \underset{=}{1} - (f_{r} o g) (λ) \end{matrix}$ That is f_ro g : (Z, {ζ}) → (Y_r, {η_r}) is syntopogenously continuous, for every r ∈ Λ.

Conversely, let each f_ro g is syntopogenously continuous, ν, ρ ∈ I^Z and (ν_i), (ρ_j) be finite families of fuzzy sets on X such that $g (ν) = \underset{i}{\lor} ν_{i}$ , $g (ρ) = \underset{j}{\land} ρ_{j}$ .

Then, by syntopogenously continuous of every f_ro g, we have $\begin{matrix} ζ (g^{\leftarrow} (ν_{i}), \underset{=}{1} - g^{\leftarrow} (ρ_{j})) \\ \geq η_{r} ((f_{r} o g) (g^{\leftarrow} (ν_{i})), \underset{=}{1} - (f_{r} o g) (g^{\leftarrow} (ρ_{j}))), \\ \forall r \in Λ \\ = η_{r} (f_{r} (ν_{i}), \underset{=}{1} - f_{r} (ρ_{j})), \\ \forall r \in Λ \end{matrix}$ That is, $\begin{matrix} ζ (g^{\leftarrow} (ν_{i}), \underset{=}{1} - g^{\leftarrow} (ρ_{j})) \geq max_{r \in Λ} \\ η_{r} (f_{r} (ν_{i}), \underset{=}{1} - f_{r} (ρ_{j})) . \end{matrix}$ Hence, $\begin{matrix} ζ (ν, \underset{=}{1} - ρ) \geq ζ (g^{\leftarrow} (\underset{i}{\lor} ν_{i}), \underset{=}{1} - g^{\leftarrow} \\ (\underset{j}{\land} ρ_{j})), by (TT 2) \\ = \underset{i, j}{\land} ζ (g^{\leftarrow} (v_{i}), \underset{=}{1} - g^{\leftarrow} (ρ_{j})), by (TT 2) \\ \geq \inf_{i, j} max_{r \in Λ} η_{r} (f_{r} (ν_{i}), \underset{=}{1} - f_{r} (ρ_{j})) . \end{matrix}$

Therefore, $ζ (ν, \underset{=}{1} - ρ) \geq ζ_{o} (g (ν), \underset{=}{1} - g (ρ))$ . Hence,g is syntopogenously continuous. □

Initial lifts in the category of T-topogenous spaces:

We denote by T-TS the category of T-topogenous spaces and their syntopogenously continuous mappings as morphisms.

In the following, we are going to show that in T-TS all initial lifts exist.

Let be a family of mappings f_r of a set X into sets Y_r, indexed by a class Λ. For each r ∈ Λ let η_r be a T-topogenous order on Y_r. For any T-topogenous order ζ on X, such that all mappings f_r : (X, {ζ}) → (Y_r, {η_r}) are syntopogenously continuous, the family (f_r : (X, {ζ}) → (Y_r, {η_r})) _r∈Λ of these mappings is called an initial lift of (f_r : X → (Y_r, {η_r})) _r∈Λ provided that for any T-topogenous space (Z, Զ), a mapping g: (Z, Զ) → (X, {ζ}) is syntopogenously continuous if and only if for all r ∈ Λ the mappings f_ro g: (Z, Զ) → (Y_r, {η_r}) are syntopogenously continuous.

Obviously, for each initial lift (f_r : (X, {ζ}) → (Y_r, {η_r})) _r∈Λ of (f_r : X → (Y_r, {η_t})) _r∈Λ, {ζ} is the coarsest T-topogenous structure on X such that each f_r : (X, {ζ}) → (Y_r, {η_r}) is syntopogenously continuous.

Now, we show that the coarsest T-topogenous structures, characterized in Theorems 2.1 and 2.2, generate canonically initial lifts.

Theorem 2.3. Let Λ be a class and for every r ∈ Λ let (Y_r, {η_r}) be a T-topogenous space and let f_r be a mapping from a set X into Y_r. Let Ք be the coarsest T-topogenous structure on X such that each f_r: (X, Ք) → (Y_r, {η_r}) is syntopogenously continuous. Then (f_r: (X, Ք) → (Y_r, {η_r})) _r∈Λ is the initial lift of (f_r: X → (Y_r, {η_r})) _r∈Λ.

Proof. Let (Z, Զ) be a T-topogenous space and g :Z → X a mapping. It is clear that, if g: (Z,Զ) → (X, Ք) is syntopogenously continuous, then by [7, Theorem 4.2] we have each mapping f_ro g: (Z, Զ) → (Y_r, {η_r}) is syntopogenously continuous. Conversely, let f_ro g be syntopogenously continuous for all r ∈ Λ and ν,ρ ∈ I^X. Then by (2) $\begin{matrix} Զ (g^{\leftarrow} (ν), g^{\leftarrow} (ρ)) \\ \geq η_{r} ((f_{r} o g) (g^{\leftarrow} (ν)), \underset{=}{1} - (f_{r} o g) (\underset{=}{1} - g^{\leftarrow} (ρ))) \\ = η_{r} ((f_{r} o g) (g^{\leftarrow} (ν)), \underset{=}{1} - (f_{r} o g) g^{\leftarrow} (\underset{=}{1} - ρ))) \\ = η_{r} (f_{r} (g (g^{\leftarrow} (ν)), \underset{=}{1} - f_{r} (g (g^{\leftarrow} (\underset{=}{1} - ρ))) \\ = η_{r} (f_{r} (ν), \underset{=}{1} - f_{r} (\underset{=}{1} - ρ)) \\ = (f_{r}^{\leftarrow} (η_{r})) (ν, ρ) \end{matrix}$ ≥ Ք(ν, ρ), because Ք is the coarsest T-topogenous order on X.

It follows, by (1) that g: (Z, Զ) → (X, Ք) is a syntopogenously continuous and this completes theproof. □

The last theorem says that all initial lifts exist uniquely in the concrete category T-TS and this means that the category T-TS is a topological [1]. Hence all initial T-topogenous structures exist. Moreover the initial T-topogenous structure of {η_r} _r∈Λ with respect to (f_r) _r∈Λ is the T-topogenous structure Ք on X which fulfills the following:

All mappings f_r: (X, Ք) → (Y_r, {η_r}) are syntopogenously continuous.

For any T-topogenous space (Z, Զ). The mapping g: (Z, Զ) → (X, Ք) is syntopogenously continuous if and only if for all r ∈ Λ the mappings f_ro g: (Z, Զ) → (Y_r, {η_r}) are syntopogenously continuous.

It is clear, by Theorem 2.3, that the coarsest T-topogenous structure, defined in Theorems 2.1 and 2.2, coincide with the initial T-topogenous structures. In other words, if f : X → Y is a mapping and η is a T-topogenous order on Y, then the initial T-topogenous structure of {η} with respect to f is Ք= f^← ({η}).

In the general case of a family ({η_r}) _r∈Λ of T-topogenous structures {η_r} on Y_r and a family (f_r) _r∈Λ of mappings f_r of a set X into Y_r, the initial T-topogenous structure of ({η_r}) _r∈Λ with respect to (f_r) _r∈Λ is given by Ք= {ζ_o}, where ζ_o as inTheorem 2.2.

In the following let us consider some special cases: The T-topogenous subspaces and the T-topogenous product spaces, in the categorical sense, are special initial T-topogenous spaces and therefore these spaces can be characterized as follows.

T-topogenous subspaces: Let E be a subset of a set Y, {η} a T-topogenous structure on Y and i: E → Y the inclusion mapping of E into Y. Then the initial T-topogenous structure of {η} with respect to i, denoted by {η_E}, will be called a T-topogenous substructure of {η} and (E, {η_E}) a T-topogenous subspace of (Y, {η}). Theorem 2.1 implies that {η_E} = {i^← (η)}.

Product T-topogenous spaces: Let ((Y_r, {η_r})) _r∈Λ be a family of T-topogenous spaces and let $X = \underset{r \in Λ}{Π} Y_{r}$ . Then the product T-topogenous structure $\underset{r \in Λ}{Π} {η_{r}}$ on X is defined to be the coarsest one of all T-topogenous orders on X such that each projection P_r : X → Y_r is syntopogenously continuous.

The pair (X, $\underset{r \in Λ}{Π} {η_{r}}$ ) is said to be a product T-topogenous space. Obviously Theorem 2.2 implies that, if Ք $= \underset{r \in Λ}{Π} {η_{r}}$ , then:

Ք(μ, λ) $= max {inf_{i, j} max_{r \in Λ} η_{r} (P_{r} (μ_{i}), \underset{=}{1} - P_{r}$ $(\underset{=}{1} - λ_{j}))}$ , where the maximum is taken over all finite families (μ_i), (λ_j) of fuzzy sets on X with $μ = \underset{i}{\lor} μ_{i}$ , $λ = \underset{j}{\land} λ_{j}$ .

A function g from a T-topogenous space (Z, Զ) to (X, Ք) is syntopogenously continuous if and only if each P_r o g: (Z, Զ) → (Y_r, {η_r}) is syntopogenously continuous.

On account of Theorem 2.2, the notion of supremum of the family {ζ_r : r ∈ Λ} of T-topogenous orders on a set is define as follows.

Theorem 2.4. Let {ζ_r : r ∈ Λ} be a family of T -topogenous orders on a set X. We define $\begin{matrix} ζ_{s} : I^{X} \times I^{X} \to I by, \\ ζ_{s} (μ, λ) = max {inf_{i, j} max_{r \in Λ} ζ_{r} (μ_{i}, λ_{j})}, \\ μ λ \in I^{X} . \end{matrix}$ where the maximum is taken over all finite families (μ_i), (λ_j) of fuzzy sets on X such that $μ = \underset{i}{\lor} μ_{i}$ , $λ = \underset{j}{\land} λ_{j}$ . Then ζ_s is a T-topogenous order on X. Moreover ζ_s is the supremum of the family {ζ_r : r ∈ Λ} and we will denote it by $sup_{r \in Λ}$ ζ_r.

Proof. We can show that ζ_s is a T-topogenous order on X, analogously as Theorem 2.2.

Now, let μ, λ ∈ I^x and (μ_i), (λ_j) be finite families of fuzzy sets on X such that $μ = \underset{i}{\lor} μ_{i}$ , $λ = \underset{j}{\land} λ_{j}$ . Then for every r ∈ Λ we have. $\begin{matrix} ζ_{s} (μ, λ) = max {inf_{i, j} max_{r \in Λ} ζ_{r} (μ_{i}, λ_{j})} \\ \geq inf_{i, j} ζ_{r} (μ_{i}, λ_{j}) \\ = ζ_{r} (\underset{i}{\lor} μ_{i}, \underset{j}{\land} λ_{j}), by (TT 2) twic \\ = ζ_{r} (μ, λ) . \end{matrix}$ This shows, by (3) that ζ_s is finer than all ζ_r.

Finally, let ζ be a T-topogenous order on X finer than all ζ_r. Then for every μ, λ ∈ I^x we have ζ^∖ (μ_i, λ_j) ≥ ζ_r (μ_i, λ_j), ∀r ∈ Λ, that is, $ζ^{∖} (μ_{i}, λ_{j}) \geq max_{r \in Λ} ζ_{r} (μ_{i}, λ_{j})$ , hence $\begin{matrix} ζ^{∖} (μ, λ) = ζ^{∖} (\underset{i}{\lor} μ_{i}, \underset{j}{\land} λ_{j}) \\ = \underset{i}{\land} \underset{j}{\land} ζ^{∖} (μ_{i}, λ_{j}), by (TT 2) twice \\ \geq inf_{i, j} max_{r \in Λ} ζ_{r} (μ_{i}, λ_{j}), by above \end{matrix}$ Therefore $\begin{matrix} ζ^{∖} (μ, λ) \geq max {inf_{i, j} max_{r \in Λ} ζ_{r} (μ_{i}, λ_{j})} \\ = ζ_{s} (μ, λ) . \end{matrix}$ This proves that ζ^∖ is finer than ζ_s.

Which completes the proof of ζ_s is the supremum of the family {ζ_r : r ∈ Λ}. □

Theorem 2.5. Let {ζ_r : r ∈ Λ} be a family of T -topogenous orders on a set X, and ζ_s be their supremum. Then $τ ({ζ_{s}}) = sup_{r \in Λ} τ ({ζ_{r}})$ .

Proof. Since ζ_r is coarser than ζ_s for every r ∈ Λ. then by Proposition 1.1, τ ({ζ_r}) ⊆ τ ({ζ_s}) for every r ∈ Λ, consequently $sup_{r \in Λ} τ ({ζ_{r}}) \subseteq τ ({ζ_{s}})$ .

On the other hand, we denote the fuzzy interior operators associated with τ ({ζ_s}) , τ ({ζ_r}) and $sup_{r \in Λ} τ ({ζ_{r}})$ , respectively, by , and . Let λ ∈ I^x, x ∈ X and (λ_j) be finite family of fuzzy sets on X such that $λ = \underset{j}{\land} λ_{j}$ . Then $\begin{array}{l} λ^{o 1} (x) = ς_{s} (1_{x}, λ) \\ = \max \{\inf_{j = 1, \dots, n} \max_{r \in λ} ς_{r} (1_{x}, λ_{j})\} \\ = \max \{\inf_{j = 1, \dots, n} \max_{r \in λ} (λ_{j}^{o 2}) (x)\} \\ \leq \max \{\inf_{j = 1, \dots, n} (λ_{j}^{03}) (x)\}, clear \\ = \max \{{(\land_{j = 1}^{n} λ_{j})}^{o 3} (x)\} \\ = λ^{o 3} (x) . \end{array}$

This establishes that $τ ({ζ_{s}}) \subseteq sup_{r \in Λ} τ ({ζ_{r}})$ and this completes the proof. □

Final lifts in the category of T-topogenous spaces:

Since the concrete category T-TS is topological [1], then all final lifts also exist and are unique. This means that also all final T-topogenous structures exist, which can be constructed as follows:

Let Λ be a class and for every r ∈ Λ let (X_r, {ζ_r}) be a T-topogenous space and g_r : X_r → Y a mapping of X_r into a set Y. The final T-topogenous structure of ({ζ_r} _{)_r∈Λ} with respect to is the T-topogenous structure {η} on Y such that all mappings g_r : (X_r, {ζ_r}) → (Y, {η}) are syntopogenously continuous and for any T-topogenous space (Z, Զ), a mapping h : (Y, η)→(Z, Զ) is syntopogenously continuous if and only if for all r ∈ Λ, the mapping h o g_r : (X_r, {ζ_r})→(Z, Զ) is syntopogenously continuous. Obviously, the final T-topogenous structure of ({ζ_r} _{)_r∈Λ} with respect to (g_r) _r∈Λ is the finest T-topogenous structure {η} on Y such that all mappings g_r : (X_r, {ζ_r}) → (Y, {η}) are syntopogenously continuous.

Quotient T-topogenous spaces: Let (X, {ζ}) be aT-topogenous space and h : X → Y a surjectivemapping.

Then the final T-topogenous structure {η} of {ζ} with respect to h will be called a quotient T-topogenous order and the pair (Y, {η}) a quotient T-topogenous space.

Sum T-topogenous spaces: Assume that for each element r of a set Λ, (X_r, {ζ_r}) is a T-topogenous space. Let $Y = \overset{+}{\underset{r \in Λ}{\cup}}$ $X_{r} = \underset{r \in Λ}{\cup} (X_{r} \times {r})$ be the disjoint union of the family (X_r) _r∈Λ and e_r : X_r → Y, for each r ∈ Λ, the related canonical injection defined by e_r (x_r) = (x_r, r). Then the final T-topogenous structure {η} of ({ζ_r}) _r∈Λ with respect to is called a sum T-topogenous structure, denoted $\overset{+}{\underset{r \in Λ}{\cup}} {ζ_{r}}$ and the pair (Y, {η}) is called a sum T-topogenous space.

Let g : X → Y be a function and ζ a T-topogenous order on X. We define the mapping g (ζ) : I^Y × I^Y → I, by for all ν, ρ ∈ I^Y : (g (ζ)) (ν, ρ) = ζ (g^← (ν) , g^← (ρ)).

We arrive the following properties

Proposition 2.1. For the mapping g (ζ) defined above, one has the following, if g is a bijective map, then g (ζ) is the finest T -topogenous order on Y such that the mapping g is syntopogenously continuous.

Proof. Obviously g (ζ) is a T-topogenous order on Y [7, Proposition 3.17].

Moreover, the syntopogenous continuity of the function g : (X, {ζ}) → (Y, {g (ζ)}) follows immediately from (1).

Now, let η be a T-topogenous order on Y and ν, ρ ∈ I^Y. Then by syntopogenous continuity of g : (X, {ζ}) → (Y, {η}), we get $\begin{matrix} (g (ζ)) (ν, ρ) & = ζ (g^{\leftarrow} (ν), g^{\leftarrow} (ρ)), \\ \geq η (v, ρ), \end{matrix}$ which shows, by (3) that g (ζ) is finer than η. This means that {g (ζ)} is the finest T-topogenous structure on Y such that g is syntopogenously continuous.

Proposition 2.2. Let ((X_r, {ζ_r} _{))_r∈Λ} be a family of T -topogenous spaces and for each r ∈ Λ, there is a bijective mapping g_r : X_r → Y. Then ${η} = {\underset{r \in Λ}{\lor} (g_{r} (ζ_{r}))}$ is the finest T -topogenous structure on Y such that each g_r : (X_r, {ζ_r}) → (Y, {η}) is syntopogenously continuous.

Proof. (i) Let (X_r, {ζ_r}) be a T-topogenous space, for all r ∈ Λ. Then obviously η satisfies (TT1) and (TT5).

(TT2) Let ν, ρ, ϑ ∈ I^Y. Then $\begin{matrix} η (ν \lor ρ, ϑ) = \underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ν \lor ρ, ϑ) \\ = \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν \lor ρ), g_{r}^{\leftarrow} (ϑ)) \\ = \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν) \lor g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ)) \\ = \underset{r \in Λ}{\lor} [ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ϑ)) \land \\ ζ_{r} (g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ))] \\ \leq [\underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ϑ))] \land \\ [\underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ))] \end{matrix}$

$\begin{matrix} = [\underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ν, ϑ)] \land \\ [\underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ρ, ϑ)] \\ = η (ν, ϑ) \land η (ρ, ϑ) . \end{matrix}$

For the converse inequality, let ɛ ∈ I₀ be such that ɛ > η (ν ∨ ρ, ϑ).

Then for every r ∈ Λ, we have $\begin{matrix} ɛ > ζ_{r} (g_{r}^{\leftarrow} (ν \lor ρ), g_{r}^{\leftarrow} (ϑ)) \\ = ζ_{r} (g_{r}^{\leftarrow} (ν) \lor g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ))) \\ = ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ϑ)) \land ζ_{r} (g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ) \end{matrix}$ that is $ɛ > ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ϑ))$ or $ɛ > ζ_{r} (g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ)), \forall r \in Λ,$ hence $\begin{matrix} ɛ \geq \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ϑ)) \\ = \underset{r \in Λ}{\lor} (g_{r} (ζ_{r}) (ν, ϑ) \\ = η (ν, ϑ)) or \\ ɛ \geq \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ρ), g_{r}^{\leftarrow} (ϑ)) \\ = \underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ρ, ϑ) \\ = η (ρ, ϑ)) \end{matrix}$ Therefore ɛ ≥ η (ν, ϑ) ∧ η (ρ, ϑ) . This clearly implies $η (ν \lor ρ, ϑ) \geq η (ν, ϑ) \land η (ρ, ϑ) .$ which proves that η (ν ∨ ρ, ϑ) = η (ν, ϑ) ∧η (ρ, ϑ). Analogously, we can show $\begin{matrix} η (ν, ρ \land ϑ) = η (ν, ρ) \land η (ν, ϑ) . \end{matrix}$ (TT3) Let ν, ρ ∈ I^Y and θ, β ∈ I₀, be such thatη(ν, ρ)>1 - (θ T β). Then there is r_o ∈ Λ such that $\begin{matrix} ζ_{ro} (g_{ro}^{\leftarrow} (ν), g_{ro}^{\leftarrow} (ρ)) & = (g_{ro} (ζ_{ro})) (ν, ρ) \\ > 1 - (θ T β) . \end{matrix}$ So, we can get C ⊆ X such that $ζ_{ro} (g_{ro}^{\leftarrow} (ν), 1_{C}) \geq 1 - θ$ and $ζ_{ro} (1_{C}, g_{ro}^{\leftarrow} (ρ)) \geq 1 - β$ .

By taking $H = g_{ro} (C) \subseteq Y (g_{ro}^{\leftarrow} (H) = C)$ , because g_ro is injective), we have $\begin{matrix} η (ν, 1_{H}) & \geq & (g_{ro} (ζ_{ro})) (ν, 1_{H}) \\ = & ζ_{ro} (g_{ro}^{\leftarrow} (ν), g_{ro}^{\leftarrow} (1_{H})) \\ = & ζ_{ro} (g_{ro}^{\leftarrow} (ν), 1_{C}) \\ \geq & 1 - θ, and \\ η (1_{H}, ρ) & \geq & (g_{ro} (ζ_{ro})) (1_{H}, ρ) \\ = & ζ_{ro} (g_{ro}^{\leftarrow} (1_{H}), g_{ro}^{\leftarrow} (ρ)) \\ = & ζ_{ro} (1_{C}, g_{ro}^{\leftarrow} (ρ)) \\ \geq & 1 - β . \end{matrix}$

(TT4) For all ν, ρ ∈ I^Y, we have $\begin{matrix} η (ν, ρ) & = & \underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ν, ρ) \\ = & \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ρ)) \\ \leq & \underset{r \in Λ}{\lor} {1 - hgt [g_{r}^{\leftarrow} (ν) T \\ (\underset{=}{1} - g_{r}^{\leftarrow} (ρ))]} \\ = & \underset{r \in Λ}{\lor} {1 - hgt [g_{r}^{\leftarrow} (ν) T g_{r}^{\leftarrow} (\underset{=}{1} - ρ)]} \\ = & \underset{r \in Λ}{\lor} {1 - hgt g_{r}^{\leftarrow} (ν T (\underset{=}{1} - ρ))} \\ = & \underset{r \in Λ}{\lor} {1 - \sup_{x \in X} [g_{r}^{\leftarrow} (ν T (\underset{=}{1} - ρ))] (x)} \\ = & \underset{r \in Λ}{\lor} {1 - \sup_{x \in X} [ν T (\underset{=}{1} - ρ)] (g_{r} (x))} \\ = & 1 - \sup_{y \in Y} [ν T (\underset{=}{1} - ρ)] (y) \\ = & 1 - hgt [ν T (\underset{=}{1} - ρ)] . \end{matrix}$ Now, let η^∖ be a T-topogenous order on Y, and ν,ρ ∈ I^Y. Then by syntopogenously continuous of each g_r : (X_r, {ζ_r}) → (Y, {η^∖}), we have $\begin{matrix} η (ν, ρ) & = & \underset{r \in Λ}{\lor} (g_{r} (ζ_{r})) (ν, ρ) \\ = & \underset{r \in Λ}{\lor} ζ_{r} (g_{r}^{\leftarrow} (ν), g_{r}^{\leftarrow} (ρ)) \\ \geq & η^{∖} (ν, ρ), \end{matrix}$ which shows, by (3) that η is finer than any T-topogenous structure η^∖ on Y such that each g_r : (X_r, {ζ_r}) → (Y, {η^∖}) is syntopogenously continuous. Therefore, {η} is the finest T-topogenous structure on Y in such a way that each g_r : (X_r, {ζ_r}) → (Y, {η}) is syntopogenouly continuous. □

2 Initial and final I -topologies

In this section, we show that the I-topology associated to the initial T-topogenous structure for a family ((Y_r, {η_r})) _r∈Λ of T-topogenous spaces and the initial I-topology for the family ((Y_r, τ ({η_r}))) _r∈Λ of I-topological spaces associated with the T-topogenous spaces coincide and we deduce an illustrative examples of this notions.

Lowen [12], introduced the concepts of initial and final I-topologies. Given a set X, an I-topological space (Y, τ) and a function f : X → Y, we can define f^← (τ) = {f^← (μ) ∈ I^X : μ ∈ τ}.

It is easy to see that f^← (τ) is the coarsest I-topology on X such that the mapping f is continuous.

On the other hand, if we consider an I-topological space (X, τ) and a function g : X → Y, then we define g (τ) = {λ ∈ I^Y : g^← (λ) ∈ τ}. It is easily seen that g (τ) is the finest I-topology on Y such that the mapping g is continuous.

More generally, consider a family of I-topological spaces and for each r ∈ Λ, a mapp g_r : X → Y_r, then it is easy to see that the union $\underset{r \in Λ}{\cup}$ $g_{r}^{\leftarrow} (τ_{r})$ is a subbase for an I-topology on X which making each g_r continuous. Moreover, it is the coarsest I-topology with this property. We denote it by $sup_{r \in Λ}$ $g_{r}^{\leftarrow} (τ_{r})$ . This I-topology is called the initial I-topology of (τ_r) _r∈Λ with respect to (g_r) _r∈Λ. On the other hand, consider a family of I-topological spaces ((X_r, τ_r)) _r∈Λ and for each r ∈ Λ, a mapping g_r : X_r → Y. It is easy to see that the intersection $\underset{r \in Λ}{\cap}$ g_r (τ_r) is the finest I-topology on Y making all the mappings g_r are continuous and fulfills the require-ments of a final lift in the category of I-topological spaces. It is called the final I-topology of (τ_r) _r∈Λ with respect to (g_r) _r∈Λ.

If E is a subset of a set Y, τ an I-topology on Y and i : E → Y the inclusion mapping of E into Y, then the initial I-topology of τ with respect to i, denoted by τ_E, will be called an I-subtopology and (E, τ_E) an I-topological subspace of (Y, τ).

Moreover, assume that for every r ∈ Λ, (Y_r, τ_r) is an I-topological space, X is the cartesian product $\underset{r \in Λ}{Π}$ Y_r of the family (Y_r) _r∈Λ and P_r : X → Y_r are the related projections. Then the initial I-topology of (τ_r) _r∈Λ with respect to the family (P_r) _r∈Λ of projections, denoted by $\underset{r \in Λ}{Π} τ_{r}$ , will be called a product I-topology and (X, $\underset{r \in Λ}{Π} τ_{r}$ ) a product I-topological space.

Proposition 3.1. [7] Let h : (X, {ζ}) → (Y, {η}) be syntopogenously continuous mapping between T -topogenous spaces. Then the mapping h : (X, τ ({ζ}))→ (Y, τ ({η})) between the associated I-topological spaces is continuous.

Proposition 3.2. Let f : X → Y be a mapping, {η} a T -topogenous structure on Y, τ ({η}) the I -topology associated to {η} and {f^← (η)} the initial T -topogenous structure of {η} with respect to f. Then the I-topology associated to {f^← (η)} coincides with the initial I-topology τ ({η}) with respect to f, that is τ ({f^← (η)}) = f^← (τ ({η})).

Proof. Since f : (X, {f^← (η)}) → (Y, {η}) is syntopogenously continuous, it follows by Proposition 3.1, that f : (X, τ ({f^← (η)})) → (Y, τ ({η})) is continuous and hence f^← (τ ({η})) ⊆ τ ({f^← (η)}).

For the opposite conclusion, we denote the fuzzy interior operators associated with τ ({f^← (η)}) and τ ({η}) respectively, by and . Let μ be τ ({f^← (η)})-open fuzzy set and x ∈ X. Then $\begin{matrix} μ (x) & = & μ^{o 1} (x) \\ = & (f^{\leftarrow} ({η})) (1_{x}, μ) \\ = & η (f (1_{x}), \underset{=}{1} - f (\underset{=}{1} - μ)) \\ = & η (f (1_{x}), λ), by putting λ = \underline{\underline{1}} - f (\underset{=}{1} - μ)) \\ = & (λ)^{o 2} (f (x)) \\ = & [f^{\leftarrow} ((λ)^{o 2})] (x) . \end{matrix}$ Thus, by above μ = f^← ((λ) ^o2) is an f^← (τ ({η}))-open fuzzy set, which clearly implies τ ({f^← (η)}) ⊆ f^← (τ ({η})). □

Let (E, η_E}) be a T-topogenous subspace of a T-topogenous space (Y, {η}) and τ ({η}) the I-topology associated to {η}. Since {η_E} is the initial T-topogenous structure of {η} with respect to the inclusion mapping i : E → Y and (τ ({η})) _E is the initial I-topology of τ ({η}) with respect to i, then from Proposition 3.2 we get directly the following result.

Corollary 3.1. The I-topology associated to {η_E} coincides with the I -subtopology (τ ({η})) _E, that is τ ({η_E}) = (τ ({η})) _E.

Proposition 3.3. Let ((Y_r, {η_r})) _r∈Λ be a family of T -topogenous spaces, τ ({η_r}) the I -topology associated to {η_r} and for every r ∈ Λ, there is a mapping f_r : X → Y_r. If Ք is the initial T -topogenous structure on X of ({η_r} _{)_r∈Λ} with respect to (f_r) _r∈Λ, then the I -topology associated to Ք coincides with the initial I -topology of the family (τ ({η_r} _{))_r∈Λ} with respect to , that is τ(Ք) $= sup_{r \in Λ} f_{r}^{\leftarrow} (τ ({η_{r}}))$ .

Proof. Since ${{f_{r}^{\leftarrow} (η_{r})} : r \in Λ}$ is a family of T-topogenous structures on the set X, then combining Theorems 2.2 and 2.4, we have $τ ({Ք}) = sup_{r \in Λ} τ ({f_{r}^{\leftarrow} (η_{r})}) = sup_{r \in Λ} f_{r}^{\leftarrow} (τ ({η_{r}})) .$

Also, by Theorem 2.5 and Proposition 3.2, we have $τ ({Ք}) = sup_{r \in Λ} τ ({f_{r}^{\leftarrow} (η_{r})}) = sup_{r \in Λ} f_{r}^{\leftarrow} (τ ({η_{r}})) .$

This completes the proof. □

The following result is a direct consequence of the last proposition.

Corollary 3.2. If $\underset{r \in Λ}{Π} {η_{r}}$ is the cartesian product of the family ({_{η
_r}} _{)_r∈Λ} of T-topogenous structures {_{η
_r}} on Y_r and Y is the cartesian product $\underset{r \in Λ}{Π}$ Y_r of the family , then the I-topology associated to $\underset{r \in Λ}{Π}$ {_{η
_r}} coincides with the product I-topology $\underset{r \in Λ}{Π} τ ({η_{r}})$ of the I-topologies τ ({η_r}) associated to {η_r}, that is $τ ({\underset{r \in Λ}{Π} {η_{r}}}) = \underset{r \in Λ}{Π} τ ({η_{r}})$ .

Proposition 3.4. Let g : X → Y be a bijective mapping, {ζ} a T-topogenous structure on X, {η} be the final T-topogenous structure of {ζ} with respect to g and τ ({ζ}), τ ({η}) the I-topologies associated with {ζ}, {η}, respectively. Then τ ({η}) coincides with the final I-topology g (τ ({ζ})) of τ ({ζ}) with respect to g.

Proof. From Proposition 3.1, it follows that g : (X, τ ({ζ})) → (Y, τ ({η})) is continuous and hence g (τ ({ζ})) is finer than τ ({η}).

Now, let ν be g (τ ({ζ}))-open fuzzy set, y ∈ Y and denote the fuzzy interior operators associated with τ ({ζ}) and τ ({η}), respectively, by and . Then ν (y) = ν (g (x)), for some x ∈ X, because g surjective $\begin{matrix} = [g^{\leftarrow} (ν)] (x) = [g^{\leftarrow} (ν)]^{o 1} (x) \\ = ζ (1_{x}, g^{\leftarrow} (ν)) \\ = ζ (g^{\leftarrow} (1_{y}), g^{\leftarrow} (ν)), because g injective \\ = (g (ζ)) (1_{y}, ν) \\ = η (1_{y}, ν) \\ = ν^{o 2} (y) . \end{matrix}$ Which implies ν is τ ({η})-open fuzzy set, thus τ (η) is finer than g (τ ({ζ})). □

Now, we define a correspondence ŧ^∼ between the category T-TS of T-topogenous spaces and the category I-TOPS of I-topological spaces, by: on objects: ŧ^∼ (X, {ζ}) = (X, τ ({ζ}))

on morphisms: ŧ^∼ is the identity function.

Then obviously, conclusion from Proposition 3.1 is that this ŧ^∼ is a well-defined functor.

Theorem 3.1. The functor ŧ^∼: T - TS → I - TOPS preserves initial lifts.

Proof. Given a source (f_r : X → (Y_r, {η_r})) _r∈Λ in T-TS, if {Ք} is the initial lift on X, then ŧ^∼(X, Ք) = (X, τ(Ք)) and from Proposition 3.3, we have $\begin{matrix} τ (Ք) = sup_{r \in Λ} f_{r}^{\leftarrow} (τ ({η_{r}})) . \end{matrix}$ Which is the initial I-topology of the family with respect to ${(f_{r})}_{r \in \land}$ . □

Now, we introduce some examples which clear the notion of initial (final) T-topogenous structures together with the corresponding initial (final) I-topology.

Example 1: Let f : X → Y be a mapping and η a T-topogenous order on Y, defined by: $\begin{matrix} η (ν, ρ) = 1 - [(hgt ν) T hgt (\underset{=}{1} - ρ)], \\ ν, ρ \in I^{Y} (see [7]) . \end{matrix}$ Then the initial T-topogenous structure of {η} with respect to f is {f^← (η)}, where, for all μ, λ ∈ I^x: $\begin{matrix} (f^{\leftarrow} (η)) (μ, λ) \\ = η (f (μ), \underline{\underline{1}} - f (\underline{\underline{1}} - λ)), by Theorem 2.1 \\ = 1 - [(hgt f (μ)) T hgt (f (\underset{=}{1} - λ))] \\ = 1 - [(hgt μ) T hgt (\underset{=}{1} - λ)], \end{matrix}$ by Lemma 1.1

Now, we observe that the effect of η on fuzzy sets of Y is the same as the effect of f^← (η) on fuzzy sets of X. So from [7. Example 3.10], we have the I-topology τ ({η}) on Y induced by η and the I-topology τ ({f^← (η)}) on X induced by f^← (η) are the indiscrete I-topologies.

Hence τ ({f^← (η)}) = f^← (τ ({η})), according to Proposition 3.2.

Example 2: Let (Y₁, {η₁}) and (Y₂, {η₂}) be two T-topogenous spaces, where η₁, η₂ defined by (see [7]): $η_{1} (ν, ρ) = 1 - hgt [ν T (\underset{=}{1} - ρ)], ν, ρ \in I^{Y_{1}};$ $η_{2} (ϑ, σ) = 1 - [(hgt ϑ) T hgt (\underset{=}{1} - σ)], ϑ, σ \in I^{Y_{2}} .$

Also, there exist mappings f₁, f₂ from a set X in to Y₁, Y₂, respectively. Then the initial T-topogenous structure of ({η_r}) _r=1, 2 with respect to (f_r) _r=1,2 is {ζ_o} given by, for all μ, λ ∈ I^x: $\begin{matrix} ζ_{o} (μ, λ) & = & max {inf_{i, j} max_{r = 1, 2} η_{r} (f_{r} (μ_{i}), \\ \underset{=}{1} - f_{r} (\underset{=}{1} - λ_{j}))}, \\ = & max {inf_{i, j} η_{1} (f_{1} (μ_{i}), \underset{=}{1} - f_{1} (\underset{=}{1} - λ_{j}))}, \\ since η_{1} \geq η_{2} \\ = & max inf_{i, j} {1 - hgt [f_{1} (μ_{i}) T \\ f_{1} (\underset{=}{1} - λ_{j})]}, \end{matrix}$ where the maximum is taken over all finite families (μ_i), (λ_j) of fuzzy sets on X such that $μ = \underset{i}{\lor} μ_{i}$ , $λ = \underset{j}{\land} λ_{j}$ .

Now, we denote the fuzzy interior operators associated with τ ({ζ_o}) and $f_{1}^{\leftarrow} (τ ({η_{i}}))$ by and , respectively. Since obviously, f₁ : (X, {ζ_o}) → (Y₁, {η₁}) is syntopogenously continuous, it is continuous from (X, τ ({ζ_o})) to (Y₁, τ ({η₁})) and hence $f_{1}^{\leftarrow} (τ ({η_{1}})) \subseteq τ ({ζ_{o}})$ .

For the opposite inclusion, let λ be τ ({ζ_o})-open fuzzy set and x ∈ X. Then put λ = ∧ _j λ_j we have $\begin{matrix} λ (x) & = & λ^{o 1} (x) \\ = & ζ_{0} (1_{x}, λ) \\ = & max inf_{j} {1 - hgt [f_{1} {(1}_{x}) T f_{1} \underset{=}{1} λ)]}, \\ = & max inf_{j} η_{1} (f_{1} (1_{x}), \underset{=}{1} - f_{1} (\underset{=}{1} - \underset{j}{Λ} λ_{j})) \\ = & max η_{1} (f_{1} (1_{x}), \underset{=}{1} - f_{1} (\underset{=}{1} - \underset{j}{Λ} λ_{j})) \\ = & η_{1} (f_{1} (1_{x}), \underset{=}{1} - f_{1} (1 - λ)) \\ = & η_{1} (f_{1} (1_{x}), v), by put ν = \underset{=}{1} - f_{1} (\underset{=}{1} - λ)) \\ = & ν^{o 2} (f_{1} (1_{x})) \\ = & [f_{1}^{\leftarrow} (v^{o 2})] (x) . \end{matrix}$ That is $λ = [f_{1}^{\leftarrow} (ν^{o 2})]$ is an $f_{1}^{\leftarrow} (τ ({η_{1}}))$ -open fuzzy set (by continuity of f₁), which clearly implies $τ ({ζ_{o}}) \subseteq f_{1}^{\leftarrow} (τ ({η_{1}}))$ and therefore, τ ({ζ_o}) = $f_{1}^{\leftarrow} (τ ({η_{1}}))$ .

Since η₁ ≥ η₂, by Proposition 1.1, we have τ ({η₁}) ⊇ τ ({η₂}). Consequently $\begin{matrix} sup_{r = 1, 2} f_{r}^{\leftarrow} (τ ({η_{r}})) = f_{1}^{\leftarrow} (τ ({η_{1}})) \\ = τ ({ζ_{o}}), \end{matrix}$ according to Proposition 3.3.

Example 3: Let g : X → Y be a bijective map and ζ a T-topogenous order on X, defined by: $\begin{matrix} ζ (μ, λ) = 1 - hgt [μ T (\underset{=}{1} - λ)], \\ μ, λ \in I^{X} (see [7]) . \end{matrix}$

Then the final T-topogenous structure of {ζ} with respect to g is {g (ζ)}, where for all ν, ρ ∈ I^Y: we have $\begin{matrix} (g (ζ)) (ν, ρ) & = & ζ (g^{\leftarrow} (ν), g^{\leftarrow} (ρ)), by Proposition 2.1 \\ = & 1 - hgt [g^{\leftarrow} (ν) T (\underset{=}{1} - g^{\leftarrow} (ρ))] \\ = & 1 - hgt [g^{\leftarrow} (ν) {Tg}^{\leftarrow} (\underset{=}{1} - ρ)] \\ = & 1 - hgt [g^{\leftarrow} (ν T (\underset{=}{1} - ρ))] \\ = & 1 - sup_{x \in X} [g^{\leftarrow} (ν T (\underset{=}{1} - ρ)] (x) \\ = & 1 - sup_{x \in X} [(ν T (\underset{=}{1} - ρ)] (g (x)) \\ = & 1 - sup_{y \in Y} [(ν T (\underset{=}{1} - ρ)] (y) \\ = & 1 - hgt [ν T (\underset{=}{1} - ρ)] . \end{matrix}$

We notice that, there is no different between the effect of ζ on the fuzzy sets of X and the effect of g (ζ) on the fuzzy sets of Y. Thus, from [7. Example 3.10], we have the I-topology τ ({ζ}) on X induced by ζ and the I-topology τ ({g (ζ)}) on Y induced by g (ζ) are the discrete I-topologies. Hence τ ({g (ζ)}) = g (τ ({ζ})), according to Proposition 3.4.

3 Conclusion

This paper, showed that the initial (final) lift and hence the initial (final) T-topogenous structures in the category of T-topogenous spaces exist. It constructed an (initial source)-preserving functor from the category of T-topogenous spaces to the category of I-topological spaces.

Footnotes

Acknowledgments

The author is grateful to an anonymous referee for his/her generous advice that significantly improved the presentation of this article.

References

10.

11.

12.

13.

14.

15.

16.

17.

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holand, Amsterdam, 1983

18.

19.