Definition 1 (cf. [33]). Let X ≠ ϕ, then $$ is called boundedness or B-set on X, if β^X satisfies the following axioms:

• (i)

  ϕ ∈ β^X

• (ii)

  B₂ ⊂ B₁ ∈ β^X implies B₂ ∈ β^X

• (iii)

  a ∈ X implies {a} ∈ β^X.

And for B-sets β^X and β^Y a function g : X⟶Y is called bounded iff it satisfies;

By BOUND we denote the corresponding defined category.

Definition 2 (cf. [33]). The triple (X, β^X, r) is called RELative space (shortly RELspace) if for the boundedness β^X the function r:$$ satisfies the following conditions:

• (i)

  B ∈ β^X and $$ implies $$

• (ii)

  {ϕ} ∉ r(B) for B ∈ β^X

• (iii)

  $$ iff $$

• (iv)

  a ∈ X implies {{a} × {a}} ∈ r({a}).

The RELspace (X, β^X, r) is called ordered-RELspace provided that the following axiom holds:

• (v)

  ϕ ≠ B₁ ⊂ B ∈ β^X implies r(B₁) ⊂ r(B).

Definition 3 (cf. [33]). Let (X, β^X, r) and (Y, β^Y, v) be two RELspaces, then a bounded function g : X⟶Y is called RELative map (shortly RELmap) iff it satisfies the following condition:

where $$ with (g × g)[R] = {(g × g)(a, c): (a, c) ∈ R} = {(g(a), g(c)): (a, c) ∈ R}. By O-REL, we denote the full subcategory of REL, whose objects are the ordered RELspaces. Note that O − REL is a bireflective subcategory of REL [34].

Example 4. Let (X, T_X) be a preuniform convergence space; then, the associated RELspace $$ can be defined as follows:

Let PU-REL denotes the category, whose objects are triples $$ and morphisms are RELmaps. Note that PUCONV≅PU-REL [9], where PUCONV is the category of preuniform convergence spaces and uniformly continuous maps as defined in [2].

Example 5. Let (X, β^X, t) be a set-convergence space; then, the associated RELspace (X, β^X, r_t) can be defined by

$$, where $$ and FIL(X) is the collection of all filters defined on X.

Let SET-REL denotes the category, whose objects are triples (X, β^X, r_t) and morphisms are RELmaps. Note that SETCONV≅SET-REL [9], where SETCONV is the category of set-convergence spaces and morphisms are b-continuous maps as defined in [3].

Example 6. Let (x, ζ) be prenearness space; then, the associated RELspace $$ can be described as

Note that PNEAR≅PN-REL [6, 9], where PNEAR is the category, whose objects are prenearness spaces and morphisms are nearness preserving maps as defined in [6], and PN-REL is the category of triples $$ and morphisms are RELmaps.

Example 7. For a B-set β^X, we put r_b(ϕ)≔{ϕ}, and for B ∈ β^X\{ϕ}, we set $$; hence, (X, β^X, r_b) defines a RELspace, which is diagonal, meaning that for B ∈ β^X\{ϕ} and $$, we can find x ∈ B such that $$.

Let Δ-REL be denote the corresponding defined full subcategory of REL; then, Δ-REL≅BOUND.

Remark 8. In this context, note that BORN, the full subcategory of BOUND, whose objects are the bornological spaces, then also has evidently a corresponding counterpart in REL.

Example 9. Let (X, β^X, q) be b-topological space; then, the associated RELspace (X, β^X, r_q) is defined by

Lemma 10. Let $$ be a collection of RELspaces. A source $$ is initial in REL iff

and for all $$,

Proof. It is given in [34]. Consequently, since O-REL is a full and isomorphism-closed subcategory which is bireflective in REL, it is topological, too.

Lemma 11. Let $$ be a collection of ordered-RELspaces. A sink $$ is final in O − REL iff

where $$, and for $$,

Proof. It is easy to observe that $$ is an ordered-RELspace and $$ is a RELmap. Suppose that $$ is a mapping. We show that g is a RELmap iff g∘f_j is a RELmap. Necessity is obvious since the composition of two RELmaps is RELmap again.

Conversely, let $$ be a RELmap.

Then, first, we show that g is a bounded map. Let $$; it implies that g(f_j(B_j)) = g∘f_j(B_j) ∈ β^Y. For our own convenience, take f_j(B_j) = B^′, and since f_j is a RELmap, then $$ and consequently, g is bounded.

Now, let $$ and $$. By the Definition 3, we have g(f_j(B_j)) = g∘f_j(B_j) ∈ r_Y(g(f_j(B_j))). On the other hand, f_j is a RELmap; it follows that $$ Take $$. Then, we have $$ and subsequently, $$ which shows g is a RELmap.

Lemma 12. Let X ≠ ϕ, and (X, β^X, r) be an ordered-RELspace.

• (i)

  A RELstructure (β^X, r) is discrete iff $$, where $$ and $$ with r_dis(ϕ)≔{ϕ}

• (ii)

  A RELstructure (β^X, r) is indiscrete iff $$, where $$ if β^X ≠ ϕ with r_id(ϕ)≔{ϕ}.

Proof. By applying Lemma 11, we get the desired result.

Remark 13. The topological functor $$, where $$ is normalized since an unique RELstructure β^X = {∅}, and r(∅) = {∅} exists whenever X = ∅ and a unique RELstructure β^X = {∅, {a}}, r(∅) = {∅} and r({a}) = {∅, {(a, a)}} exists whenever X = {a}. Furthermore, the topological functor $$ is geometric since the regular sub-object of a discrete RELspace is discrete, and finite product of discrete RELstructures is discrete again.

Definition 14 (cf. [14]).

• (i)

  A mapping A_p : X∨_pX⟶X² is said to be principalp-axis mapping provided that

• (ii)

  A mapping S_p : X∨_pX⟶X² is said to be skewedp-axis mapping provided that

• (iii)

  A mapping ∇_p : X∨_pX⟶X is said to be fold mapping atp provided that

Assume that $$ is a topological functor, $$ with $$ and p ∈ Z.

Definition 15 (cf. [14]).

• (i)

  X is $$ at p provided that the initial lift of the $$-source $$ is discrete

• (ii)

  X is $$ at p provided that the initial lift of the $$-source $$ is discrete, where X∨_pX is the wedge product in $$, i.e., the final lift of the $$-sink $$, where i₁, i₂ represent the canonical injections

• (iii)

  X is T₁ at p provided that the initial lift of the $$-source $$ is discrete.

Remark 16.

• (i)

  In TOP, $$ and T′₀ at p (respectively, T₁ at p) are equivalent to the classical T₀ at p (respectively, the classical T₁ at p), i.e., for each a ∈ X with a ≠ p, there exists a neighborhood N_a of  ^{``</sup>a<sup>"</sup> not containing  <sup>``}p^" or (respectively, and); there exists a neighborhood N_p of  ^{``</sup>p<sup>"</sup> not containing  <sup>``}a^" [35]

• (ii)

  A topological space X is T₀ (respectively T₁) iff X is T₀ (respectively T₁) at p for each p ∈ X [35]

• (iii)

  Let $$ be a topological functor, $$ and $$ be a retract of X. Then, if X is $$ or T₁ at p, then X is $$ at p but not conversely in general [36].

Theorem 17. Let (X, β^X, r) be ordered-RELspace and p ∈ X. Then, (X, β^X, r) is $$ at p if and only if for each a ∈ X with a ≠ p, the following holds:

• (i)

  {a, p} ∉ β^X

• (ii)

  $$

• (iii)

  $$ or $$

• (iv)

  $$ or $$

Proof. Let (X, β^X, r) be $$ at p; we show the conditions (i) to (iv) are holding:

• (i)

  Suppose that {a, p} ∈ β^X for all a ∈ X with a ≠ p. Let U = {a₁, a₂} ∈ X∨_pX, then since $$ and for j = 1, 2, π_jA_p(U) = {a, p} ∈ β^X (by the assumption), where π_j : X²⟶X for j =1,2 are projection maps. By Definitions 1 and 15 and Lemma 10, a contradiction, it follows {a, p} ∉ β^X

• (ii)

  Assume that $$ and $$. Particularly, let $$ and $$; then, $$. By the assumption, $$ and $$. Since (X, β^X, r) is $$ at p, it follows that $$, where $$ is the discrete structure on X∨_pX.

Similarly, for $$, we get $$, a contradiction to the discreteness of $$.

Thus, $$ or $$.

• (iii)

  Suppose that $$ and $$. In particular, let $$ and $$; then, $$, and by the assumption $$ and $$. Since (X, β^X, r) is $$ at p, we get that $$, where $$ is the discrete structure on X∨_pX

Similarly, for $$, we get $$, a contradiction.

Therefore, $$ or $$.

• (iv)

  Assume that $$ and $$. Let $$ and $$; then, $$, $$, $$ (by the assumption). Since (X, β^X, r) is $$ at p, it follows that $$, where $$ is the discrete structure on X∨_pX.

Similarly, for $$, we get $$, a contradiction.

Hence, $$ or $$.

Conversely, suppose (i) to (iv) are holding.

Let $$ be the initial structure induced by $$ and $$, where $$ is the product RELstructure on X² and $$ the discrete RELstructure on X.

We show that $$ is the discrete REL structure on X∨_pX, i.e., we show that $$ and for $$, $$.

Let $$ and $$; if ∇_pU = ∅, then U = ∅. Suppose ∇_pU ≠ ∅. Then, we have ∇_pU = {a} for some a ∈ X, and if a = p, then U = {p}; let a ≠ p; then, it further implies that U = {a₁} or U = {a₂} and U = {a₁, a₂}. By the assumption, π_jA_pU = π_jA_p{a₁, a₂} = {a, p} ∉ β^X (for j =1,2). Thus, U = {a₁} and U = {a₂}; subsequently, $$.

Now, $$ implies B = {a₁} and B = {a₂}, and by Lemma 10, $$ = $$

Suppose B = {a₁}, then

$$; it follows that $$.

Since $$, we have the following possibilities of $$:

$$,

$$,

$$,

$$,

$$.

•  

  Case (i). Suppose $$. It follows that for all $$ such that $$, $$. By Definition 2, $$. Similarly, $$. Therefore, $$ holds

•  

  Case (ii). $$ holds. The proof is similar to Case (i)

•  

  Case (iii). Let $$. It follows that for all $$ such that $$, and $$, $$. By the assumption, we get $$. Similarly, $$. Thus, $$ cannot be possible

•  

  Case (iv). Similar to Case (iii), we conclude that $$ is not possible

•  

  Case (v). If $$. It follows that for all $$ such that $$, and $$, for all $$ implying $$. By the assumption, $$. Similarly, $$. Hence, $$ is not possible.

Theorem 18. Let (X, β^X, r) be an ordered-RELspace and p ∈ X.

(X, β^X, r) is T₁ at p if and only if for any a ∈ X with a ≠ p, the following holds:

• (i)

  {a, p} ∉ β^X

• (ii)

  $$ and $$

• (iii)

  $$ and $$

• (iv)

  $$ and $$.

Proof. By following the same technique used in Theorem 17, and replacing the mapping A_p by the mapping S_p, we get the proof.

Theorem 19. All ordered-RELspaces are T′₀ at p.

Proof. Let (X, β^X, r) be ordered-RELspace and p ∈ X. By Definition 15, we show that for each $$ (where k = 1, 2) for some V ∈ β^X and $$. ∇_pU = ϕ implying U = ϕ. Suppose ∇_pU ≠ ϕ, it implies that ∇_pU = {a} for some a ∈ X. If a = p, then ∇_pU = {p} implying U = {p}.

Suppose a ≠ p, it follows that U = {a₁}, {a₂} or {a₁, a₂}. If U = {a₁, a₂}, then {a₁, a₂} ⊂ i₁(V) for some V ∈ β^X which shows that a₂ should be in the first component of the wedge product X∨_pX, a contradiction. In similar manner, $$ for some V ∈ β^X. Hence, $$. Thus, we must have U = {a_j} for j = 1, 2 only and consequently, $$, the discrete RELstructure on X∨_pX.

Now, for $$, by Lemma 10, $$ = $$. Since $$, we have the following possibilities of $$:

$$,

$$,

$$,

$$,

$$.

In particular, for $$. It follows that, for all $$ such that $$, and (for k =1,2), $$ implying $$. It follows a₂ (respectively, a₁) in the first (respectively, second) component of the wedge product X∨_pX, a contradiction. Similarly, for $$ and $$, we get a contradiction.

Therefore, $$. Consequently, by Definition 15(i) and Lemma 10, (X, β^X, r) is T′₀ at p.

Definition 20 (cf. [14]).

• (i)

  A mapping A : X²∨_ΔX²⟶X³ is called principal axis mapping provided that

• (ii)

  A mapping S : X²∨_ΔX²⟶X³ is called skewed axis mapping provided that

• (iii)

  A mapping ∇:X²∨_ΔX²⟶X² is called fold mapping provided that

Definition 21. Let $$ be a topological functor, $$ with $$.

• (i)

  X is $$ provided that the initial lift of the $$-source $$ is discrete [14]

• (ii)

  X is T′₀ provided that the initial lift of the $$-source $$ is discrete, where $$ is the final lift of the $$-sink $$ [14, 39]

• (iii)

  X is called T₀ provided that X doesn’t contain an indiscrete subspace with at least two points [18, 40]

• (iv)

  X is T₁ provided that the initial lift of the $$-source $$ is discrete [14].

Remark 22.

• (i)

  In TOP, all the properties of being T₀, $$ and $$ (respectively, T₁) are equivalent to those classical ones which are T₀ (respectively, T₁), i.e., for each a, b ∈ X with a ≠ b, there exists a neighbourhood N_a of  ^{``</sup>a<sup>"</sup> not containing  <sup>``}b^" or (respectively and), there exists a neighbourhood N_b of  ^{``</sup>b<sup>"</sup> not containing  <sup>``}a^" [14, 18, 40]

• (ii)

  In any topological category, $$ implies is T′₀ but not conversely in general. Also, each of the $$ and T′₀ has no relation to a T₀ [39]

• (iii)

  Let $$ be a topological functor, $$ and $$ be a retract of X. Then, if X is $$ (respectively T₁), then X is $$ at p (respectively T₁ at p) but not conversely in general [36].

Theorem 23. Let (X, β^X, r) be an ordered-RELspace.

(X, β^X, r) is $$ iff for each a, b ∈ X with a ≠ b, the following holds:

• (i)

  {a, b} ∉ β^X

• (ii)

  $$ or $$

• (iii)

  $$ or $$

• (iv)

  $$ or $$.

Proof. Suppose (X, β^X, r) is $$, we show that conditions (i) to (iv) are holding.

• (i)

  Suppose that {a, b} ∈ β^X for each a,b∈ X, a ≠ b. Let U = {((a, b)₁, (a, b)₂)} ∈ X²∨_ΔX². Note that $$ and π₁A(U) = {a} ∈ β^X. By the assumption, π_kA(U) = π_kA{((a, b)₁, (a, b)₂)} = {a, b} ∈ β^X, where π_k : X³⟶X² (for k =2,3) are projection maps. By Definitions 1 and 15 and Lemma 10, it leads to a contradiction, it follows that {a, b} ∉ β^X

• (ii)

  Suppose that $$ and $$. Let $$ and $$, then $$. By Definition 2, π₁A{{((a, b)₁, (a, b)₂)}} = {{(π₁A(a, b)₁, π₁A(a, b)₂)}} = {{(a, a)}} ∈ r({a}) and by the assumption, π₂A{{((a, b)₁, (a, b)₂)}} = {{(b, a)}} ∈ r({b}) and π₃A{{((a, b)₁, (a, b)₂)}} = {{(a, b)}} ∈ r({a}). Since (X, β^X, r) is $$, we conclude $$, where $$ is the discrete structure on X²∨_ΔX²

Similarly, for $$, we get $$, a contradiction.

Therefore, $$ or $$.

• (iii)

  Suppose that $$ and $$. In particular, let $$ and $$, then $$. By Definition 2, π₁A{{((a, b)₂, (a, b)₁)}} = {{(a, a)}} ∈ r({a}) and by the assumption, π₂A{{((a, b)₂, (a, b)₁)}} = {{(π₂A(a, b)₂, π₂A(a, b)₁)}} = {{(a, b)}} ∈ r({b}) and π₃A{{((a, b)₂, (a, b)₁)}} = {{(π₃A(a, b)₂, π₃A(a, b)₁)}} = {{(b, a)}} ∈ r({a}). Since (X, β^X, r) is $$ it follows that $$, where $$ is the discrete structure on X²∨_ΔX²

Similarly, for $$, we get $$, a contradiction.

Thus, $$ or $$.

• (iv)

  Suppose that $$ and $$. Let $$, and $$; then, $$, and by Definition 2, π₁A{{(((a, b)₁, (a, b)₁), ((a, b)₂, (a, b)₂)}} = {{(a, a)}} ∈ r({a}). By the assumption, π₂A{{(((a, b)₁, (a, b)₁), ((a, b)₂, (a, b)₂)}} = {{(b, b), (a, a)}} ∈ r({b}) and π₃A{(((a, b)₁, (a, b)₁), ((a, b)₂, (a, b)₂)}} = {{(a, a), (b, b)}} ∈ r({a}). Since (X, β^X, r) is $$, we conclude $$, where $$ is the discrete structure on X²∨_ΔX².

Similarly, for $$, we get $$, a contradiction to the discreteness of $$.

Hence, $$ or $$.

Theorem 24. Let (X, β^X, r) be an ordered-RELspace.

(X, β^X, r) is T₀ iff for each a, b ∈ X with a ≠ b, each of the following conditions are satisfied:

• (i)

  {a, b} ∉ β^X

• (ii)

  $$ or $$

• (iii)

  $$ or $$

• (iv)

  $$ or $$.

Proof. Let (X, β^X, r) be T₀, {a, b} ∈ β^X and $$ and $$ and $$ and $$ and $$ and $$.

Let U = {a, b}. Note that (U, β^U, r_U) is the subspace of (X, β^X, r), where (β^U, r_U) is the initial lift of the ordered-RELsystem induced by the inclusion map i : S⟶U and for any S ⊂ U, S ∈ β^U, whenever i(S) = S ∈ β^U and for any $$, whenever $$.

By the assumption, i(U) = U = {a, b} ∈ β^U and by Definition 1, we get $$.

Now, for any $$ let $$. By Definition 2, i({{(a, a)}}) = {{(a, a)}} ∈ r({a}). By the assumption, $$ implying that $$ and $$.

Similarly, for $$, it follows that $$ and $$.

Now, if $$ then by the assumption, $$ and $$.

And for $$ then by the assumption, $$ and $$.

Therefore, $$ and $$, which is a contradiction by Lemma 12. Thus (i) − (iv) are holding.

Conversely, suppose that for all a, b ∈ X with a ≠ b, conditions (i) − (iv) are holding. We show that the initial structure (β^U, r_U) is not an indiscrete ordered-RELstructure on U. Let U = {a, b} ⊂ X. By the assumption, {a, b} ∉ β^X and $$ or $$ and $$ or $$ and $$ or $$. Thus, (U, β^U, r) is not an indiscrete ordered-RELsubspace of (X, β^X, r). Hence, by Definition 21 (iii), (X, β^X, r) is T₀.

Theorem 25. Let (X, β^X, r) be an ordered-RELspace. Then, (X, β^X, r) is T₁ iff for all a, b∈ X with a ≠ b, the following holds:

• (i)

  {a, b} ∉ β^X

• (ii)

  $$ and $$

• (iii)

  $$ and $$

• (iv)

  $$ and $$.

Proof. Similarly, using Theorem 23, and replacing mapping A by the mapping S, we obtain the proof.

Theorem 26. All ordered-RELspaces are T′₀.

Proof. Let (X, β^X, r) be an ordered-RELspace. By Definition 21, we show that for any $$, U ⊂ i_k(V) (where k = 1, 2) for some $$ and $$. If ∇U = ϕ implies U = ϕ. Suppose ∇U ≠ ϕ, hence ∇U = {(a, b)} for some (a, b) ∈ X².

Suppose a ≠ b, it follows that U = {(a, b)₁}or{(a, b)₂}or{(a, b)₁, (a, b)₂}. If U = {(a, b)₁, (a, b)₂}, then {(a, b)₁, (a, b)₂} ⊂ i₁(V) for some $$, which shows that (a, b)₂ must be in the first component of X²∨_ΔX², a contradiction. Similarly, $$, for $$. Hence, U = {{(a, b)_j}} for j = 1, 2. Consequently, $$, the discrete ordered-RELstructure on X²∨_ΔX².

Now, for $$, and by Lemma 10, $$. But $$ gives the following possibilities:

$$,

$$,

$$,

$$,

$$.

In particular, for $$. Then, it follows, for all $$$$, and consequently, $$ (for k =1,2). As a result, (a, b)₂ (respectively, (a, b)₁) is in the first (respectively, second) component of the wedge product X²∨_ΔX² which leads to a contradiction. Similarly, for $$ and $$ we get a contradiction.

Hence, $$. Thus, by Lemma 10 and Definition 21, (X, β^X, r) is T′₀.

Remark 27. Let X be an ordered-RELspace.

• (i)

  By Theorems 17 and 23, X is $$ iff X is $$ at p, for each p ∈ X

• (ii)

  By Theorems 18 and 25, X is T₁ iff X is T₁ at p, for each p ∈ X

• (iii)

  By Theorems 19 and 26, X is T′₀ iff X is T′₀ at p, for each p ∈ X

• (iv)

  By Theorems 23–26, T₁⟹$$T₀⟹T′₀ but the converse does not hold in general.

Corollary 28. Let $$ be in PU − REL. Then the following statements are equivalent.

• (i)

  $$ is $$

• (ii)

  $$ is $$, where $$ is the category of $$ pre-uniform convergence spaces and uniformly continuous maps

• (iii)

  For each a, b ∈ X with $$, and for all $$, $$ or $$, and $$.

Proof. By applying Example 4, Theorem 23, and Theorem 3.1.10 of [41].

Corollary 29. Let $$ be in PU − REL. Then, the following statements are equivalent:

• (i)

  $$ is T₁

• (ii)

  $$ is T₁PUCONV, where T₁PUCONV is the category of T₁ pre-uniform convergence spaces and uniformly continuous maps

• (iii)

  For all a, b ∈ X with $$, and for all $$, $$ and $$, and $$.

Proof. This follows from Example 4, Theorem 25, and Theorem 3.2.4 of [41].

Definition 30 (cf. [42]). Given a topological functor $$, and a full and isomorphism-closed subcategory $$ of $$, we say that $$ is

• (i)

  Epireflective in $$ and closed if and only if $$ is closed under the formation of products and extremal subobjects (i.e., subspaces)

• (ii)

  Quotient-reflective in $$ if and only if $$ is epireflective and is closed under finer structures (i.e., if $$, $$, $$, and id : A⟶B is a $$-morphism, then $$).

Theorem 31.

• (i)

  Any $$O-REL, T₀O-REL and T₁O-REL is a quotient-reflective subcategory of O-REL

• (ii)

  $$O-REL is a normalized topological construct

Proof. (i) Suppose $$$$O − REL and $$. It can be easily verified that $$ is a full and isomorphism-closed subcategory of O-REL and closed under finer structures. It remains to show that X is closed under extremal subobjects and closed under the formation of products.

Let A ⊂ X and (β^A, r_A) denotes the sub O-REL structure on A, induced by the inclusion map i : A⟶X. We show that (A, β^A, r_A) is $$O-REL space. Suppose that for any a,b ∈ A with a ≠ b, {a, b} ∈ β^A, then by the inclusion map i({a, b}) = {i(a), i(b)} = {a, b} ∈ β^X, a contradiction by Theorem 23.Thus, {a, b} ∉ β^A.

Now, suppose $$ and $$. It follows that, for all $$ such that {(a, b)} ⊂ R, and by the inclusion map i{(a, b)} ⊂ i(R) implying {(a, b)} ⊂ R. It follows that $$, a contradiction by Theorem 23. Similarly, by the same argument $$, a contradiction. Therefore, $$ or $$.

In similar way, $$ or $$, and $$ or $$. Hence, X is closed under extremal subobjects.

Next, suppose that X = ∏_k∈IX_k, where $$ are the $$O-REL structures on X_k induced by projection map π_k : X_k⟶X for all k ∈ I, i.e., $$. We show that (X, β^X, r_X) is a $$O-REL space. Let {a, b} ∈ β^X for any a, b ∈ X with a ≠ b. Then, $$, a contradiction by Theorem 23. Thus, {a, b} ∉ β^X.

Now, suppose $$ and $$. It follows $$ implies {(a, b)} ⊂ R. Then, there is k ∈ I for which a_k ≠ b_k ∈ X_k, and π_k{(a, b)} ⊂ π_kR implying {(π_ka, π_kb)} = {(a_k, b_k)} ⊂ π_kR. It follows that $$, a contradiction by Theorem 23. By the same process, $$, a contradiction. Hence, $$ or $$. In similar way, $$ or $$, and $$ or $$. Hence, X is closed under the formation of products.

Therefore, the category $$O-REL is a quotient-reflective subcategory of O-REL.

Analogous to the above argument, setting $$O − REL or T₁O − REL, the proof can be easily followed by using Theorem 24 or Theorem 25, respectively.

(ii) By the Theorem 26 and Remark 13, $$O-REL and O-REL are isomorphic categories and thus $$O-REL is normalized