This study is primarily concerned with the following concept.
Definition 3.1
An STS (X, τ, A) is called soft ω^{*}-paracompact if for every ℋ ∈ SOC (X, τ_{ω}, A), there is ℳ ∈ SOC (X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
The following three Lemmas are used in Theorem 3.6 below.
Lemma 3.2
If ℋ is soft locally finite in (X, τ, A), then {Cl_{τ}_{ω} (H) : H ∈ ℋ} is soft locally finite in (X, τ, A).
Proof
Let a_{x} ∈ SP(X, A); then, there is F ∈ τ such that a_{x} ∈ F, and {H ∈ ℋ: F∩̃H ≠ 0_{A}} is finite.
Claim
{H ∈ ℋ: F∩̃Cl_{τ}_{ω} (H) ≠ 0_{A}} ⊆ {H ∈ ℋ: F∩̃H ≠ 0_{A}}.
Proof of Claim
We assume that F ∩̃ Cl_{τ}_{ω} (H) ≠ 0_{A} for some H ∈ ℋ. We select a_{x} ∊̃ F ∩̃ Cl_{τ}_{ω} (H) ⊂̃ F ∩̃ Cl_{τ} (H); then, a_{x} ∊̃ F ∩̃ Cl_{τ} (H), and thus F ∩̃ H ≠ 0_{A}.
Lemma 3.3
If ℋ is soft locally finite in (X, τ, A), then ℋ is soft locally finite in (X, τ_{ω}, A).
Proof
Let a_{x} ∈ SP(X, A); then, there is F ∈ τ ⊆ τ_{ω} such that a_{x} ∈ F, and {H ∈ ℋ: F∩̃H ≠ 0_{A}} is finite.
The following example shows that the converse of the implication in Lemma 3.3 is not true in general.
Example 3.4
Let X = ℕ, τ = {0_{A}, 1_{A}}, A = ℝ, and ℋ = SP(X, A). Then, ℋ is soft locally finite in (X, τ_{ω}, A), but it is not soft locally finite in (X, τ, A).
Lemma 3.5
If ℋ is soft locally finite in (X, τ, A), then
$$\tilde{\cup}\{{Cl}_{{\tau}_{\omega}}\hspace{0.17em}(H):H\in \mathscr{H}\}={Cl}_{{\tau}_{\omega}}\hspace{0.17em}(\tilde{\cup}\{H:H\in \mathscr{H}\}).$$
Proof
We assume that ℋ is soft locally finite in (X, τ, A). By Lemma 3.3, ℋ is soft locally finite in (X, τ_{ω}, A). Thus, by applying Proposition 5.2 in [37], we obtain the result.
The following provides two characterizations of soft ω^{*}-paracompact STSs.
Theorem 3.6
Let (X, τ, A) be an STS so that (X, τ_{ω}, A) is soft regular. Then, the following are equivalent:
(a) (X, τ, A) is soft ω^{*}-paracompact.
(b) For every ℋ ∈ SOC(X, τ_{ω}, A), there is ℳ ∈ SC(X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
(c) For every ℋ ∈ SOC(X, τ_{ω}, A), there is ℳ ∈ SCC(X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
Proof
(a) ⇒ (b): Obvious.
(b) ⇒ (c): We assume that (X, τ, A) is soft ω^{*}-paracompact, and let ℋ ∈ SOC (X, τ_{ω}, A). For every a_{x} ∈ SP(X, A), we select H_{a}_{x} ∈ ℋ such that a_{x} ∊̃ H_{a}_{x}. As (X, τ_{ω}, A) is soft regular, by Proposition 2.7, for every a_{x} ∈ SP(X, A), there is N_{a}_{x} ∈ τ such that a_{x} ∊̃ N_{a}_{x} ⊂̃ Cl_{τ}_{ω} (N_{a}_{x}) ⊂̃ H_{a}_{x}. Let ; then, . As (X, τ, A) is soft ω^{*}-paracompact, there is ℳ ∈ SOC(X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and . Letℳ_{1} = {Cl_{τ}_{ω} (M) : M ∈ ℳ}. Then, ℳ_{1} ∈ SCC(X, τ_{ω}, A), and by Lemma 3.2, ℳ_{1} is soft locally finite.
Claim (i)
ℳ_{1} ⪯̃ ℋ.
Proof of Claim
Let Cl_{τ}_{ω} (M) ∈ ℳ_{1}, where M ∈ ℳ. As , there is a_{x} ∈ SP(X, A) such that M ⊂̃ N_{a}_{x}, and thus Cl_{τ}_{ω} (M) ⊂̃ Cl_{τ}_{ω} (N_{a}_{x}) ⊂̃ H_{a}_{x} with H_{a}_{x} ∈ ℋ. It follows thatℳ_{1} ⪯̃ ℋ.
(c) ⇒ (a): Let ℋ ∈ SOC (X, τ_{ω}, A). By (c), there is ℒ ∈ SCC (X, τ_{ω}, A) such that ℒ is soft locally finite in (X, τ, A) and ℒ ⪯̃ ℋ. As ℒ is soft locally finite, for each a_{x} ∈ SP(X, A), there is N_{a}_{x} ∈ τ such that a_{x} ∊̃ N_{a}_{x}, and {L ∈ ℒ: N_{a}_{x} ∩̃L ≠ 0_{A}} is finite. Let . Then, , and by (c), there is such that is soft locally finite in (X, τ, A) and . As ℒ ⪯̃ ℋ, for every L ∈ ℒ, there is H_{L} ∈ ℋ such that L ⊂̃ H_{L}. For every L ∈ ℒ, let and M_{L} = K_{L} ∩̃ H_{L}.
Claim (ii)
(1) {K_{L} : L ∈ ℒ}. ⊆ τ_{ω}.
(2) For every L ∈ ℒ, we have L ⊆̃ K_{L}.
(3) For every L ∈ ℒ and , K_{L} ∩̃ G ≠ 0_{A} if and only if L ∩̃ G ≠ 0_{A}.
(4) {K_{L} : L ∈ ℒ} is soft locally finite.
(5) {M_{L} : L ∈ ℒ} ∈ SOC (X, τ_{ω}, A).
(6) {M_{L} : L ∈ ℒ} ⪯̃ ℋ.
(7) {M_{L} : L ∈ ℒ} is soft locally finite.
Proof of Claim
1. Follows from Lemma 3.5.
2. We assume toward a contradiction that there are L_{0} ∈ ℒ and a_{x} ∈ SP(X, τ) such that . Then, a_{x} ∊̃ L_{0} and . As , there is such that G_{x} ∩̃ L_{0} = 0_{A} and a_{x} ∊̃ G_{x}. However, a_{x} ∊̃ G_{x} ∩̃ L_{0}, which is a contradiction.
3. Necessity. Let L ∈ ℒ and such that K_{L} ∩̃ G ≠ 0_{A}, and we assume toward a contradiction that L ∩̃ G = 0_{A}. We select a_{x} ∊̃ K_{L} ∩̃ G. Then, a_{x} ∊̃ K_{L} and a_{x} ∊̃ G. As L ∩̃ G = 0_{A} and a_{x} ∊̃ G, we have and so a_{x} ∉̃ K_{L}, which is a contradiction.
Sufficiency
Let L ∈ ℒ and such that L ∩̃ G ≠ 0_{A}. By (2), L ⊂̃ K_{L}, and therefore L ∩̃ G ⊂̃ K_{L} ∩̃ G. It follows that K_{L} ∩̃ G ≠ 0_{A}.
4. Let a_{x} ∈ SP(X, τ). As is soft locally finite, there is S ∈ τ such that a_{x} ∊̃ S, and { : G∩̃S ≠ 0_{A}} is finite; let { : G∩̃S ≠ 0_{A}} = {G_{1}, G_{2}, …, G_{n}}. We assume that for some L ∈ ℒ, we have K_{L} ∩̃ S ≠ 0_{A}. We select b_{y} ∊̃ K_{L} ∩̃ S. As b_{y} ∊̃ 1_{A} = ∪̃ {G : }, there is such that b_{y} ∊̃ G. As b_{y} ∊̃ K_{L}, by (3), we have G ∩̃ L ≠ 0_{A}. As b_{y} ∊̃ G ∩̃ S, we have G = G_{i} for some i = 1, 2, …, n. As , for every i = 1, 2, …, n, there is (a_{i})_{x}_{i} such that G_{i} ⊂̃ N_{(}_{ai})_{xi}. Therefore, {L ∈ ℒ : K_{L}∩̃S ≠ 0_{A}} is finite. It follows that {K_{L} : L ∈ ℒ} is soft locally finite.
5. Let L ∈ ℒ. As H_{L} ∈ ℋ ⊆ τ_{ω}, and by (1), we have K_{L} ∈ τ_{ω}, it follows that M_{L} = K_{L} ∩̃ H_{L} ∈ τ_{ω}. However, by (2) L. Moreover, L ⊂̃ H_{L}, and by (2), we have L ⊂̃ K_{L}; accordingly, L ⊂̃ M_{L}. It follows that {M_{L} : L ∈ ℒ} ⊆ τ_{ω} and 1_{A} = ∪̃ {L : L ∈ ℒ} ⊂̃ ∪̃ {M_{L} : L ∈ ℒ}. Therefore, {M_{L} : L ∈ ℒ} ∈ SOC(X, τ_{ω}, A).
6. Let L ∈ ℒ. Then, M_{L} = K_{L} ∩̃ H_{L} ⊂̃ H_{L} with H_{L} ∈ ℋ. Thus, {M_{L} : L ∈ ℒ} ⪯̃ ℋ.
7. This follows directly from (4).
The following characterization of soft paracompact STSs is used in the proof of Theorem 3.8.
Proposition 3.7
Let (X, τ, A) be an STS, and let ℬ be a soft base of τ. Then, (X, τ, A) is soft paracompact if and only if for every ℋ ∈ SOC(X, τ, A) with ℋ ⊆ ℬ, there is ℳ ∈ SOC (X, τ, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
Theorem 3.8
Let X be an initial universe, and let A be a set of parameters. Moreover, let {ℑ_{a} : a ∈ A} be an indexed family of topologies on X. Then, $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact if and only if (X, ℑ_{a}) is paracompact for all a ∈ A.
Proof. Necessity
We assume that $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact. Let b ∈ A. To show that (X, ℑ_{b}) is paracompact, let . We set . Then, $\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$. As $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft paracompact, there is $\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ such that ℳ is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ and ℳ ⪯̃ ℋ. Let ℳ_{b} = {M(b) : M ∈ ℳ}.
Claim
(1) ℳ_{b} ∈ OC(X, ℑ_{b}).
(2) ℳ_{b} is locally finite in (X, ℑ_{b}).
(3) .
Proof of Claim
1. As $\mathcal{M}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$, by the definition of ℑ_{b}, we have ℳ_{b} ⊆ ℑ_{b}. As $\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$, we have ∪̃{M : M ∈ ℳ} = 1_{A}, and thus (∪̃{M : M ∈ ℳ}) (b) = ∪{M(b) : M ∈ ℳ} = X. Therefore, ℳ_{b} ∈ OC(X, ℑ_{b}).
2. Let x ∈ X. Then, b_{x} ∈ SP(X, A). As ℳ is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$, there is $F\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$such that b_{x} ∊̃ F, and {M ∈ ℳ: F∩̃M ≠ 0_{A}} is finite. Thus, x ∈ F(b) ∈ ℑ_{b}. If M ∈ ℳ is such that F(b) ∩ M(b) ≠ ∅︀, then F ∩̃ M ≠ 0_{A}. It follows that ℳ_{b} is locally finite in (X, ℑ_{b}).
3. Let M(b) ∈ ℳ_{b} − {∅︀}, where M ∈ ℳ. As ℳ ⪯̃ ℋ, there is H ∈ ℋ such that M ⊂̃ H, and thus M(b) ⊆ H(b). As M(b) ≠ ∅︀, we have H(b) ≠ ∅︀, and thus there is such that H = b_{U} and H(b) = U. It follows that .
Sufficiency
We assume that (X, ℑ_{a}) is paracompact for all a ∈ A. Let ℬ = {a_{Y} : a ∈ A and Y ∈ ℑ_{a}}. By Theorem 3.5 in [1], ℬ is a soft base of $\underset{a\in A}{\oplus}}{\Im}_{a$. We apply Proposition 3.7. Let $\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ with ℋ ⊆ ℬ. For each a ∈ A, let ℋ_{a} = {Y ⊆ X : a_{Y} ∈ ℋ}. Then, for all a ∈ A, ℋ_{a} ∈ OC (X, ℑ_{a}), where (X, ℑ_{a}) is paracompact, and thus there is ℳ_{a} ∈ OC (X, ℑ_{a}) such that ℳ_{a} is locally finite in (X, ℑ_{a}) and ℳ_{a} ⪯ ℋ_{a}. Let .
Claim
(1) $\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
(2) is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
(3) .
Proof of Claim
1. For all a ∈ A, we have ℳ_{a} ⊆ ℑ_{a}, and therefore $\mathcal{G}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$. For all a ∈ A, we have ℳ_{a} ∈ OC (X, ℑ_{a}), and thus (∪̃{a_{Y} : a ∈ A and Y ∈ ℳ_{a}}) (a) = ∪{Y : Y ∈ ℳ_{a}} = X. Therefore, ∪̃{a_{Y} : a ∈ A and Y ∈ ℳ_{a}} = 1_{A}. It follows that $\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
2. Let b_{x} ∈ SP(X, A). As ℳ_{a} is locally finite in (X, ℑ_{b}), there is O ∈ ℑ_{b} such that x ∈ O, and {Y : O ∩ Y ≠ ∅︀} is finite.
We have ${b}_{x}\in {b}_{O}\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$. If b_{O} ∩̃ a_{Y} ≠ 0_{A}, then a = b and O ∩ Y ≠ ∅︀. This shows that {a_{Y} : a ∈ A and Y ∈ ℳ_{a}, and b_{O}∩̃a_{Y} ≠ 0_{A}} is finite. It follows that is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
3. Let with a ∈ A and Y ∈ ℳ_{a}. As ℳ_{a} ⪯ ℋ_{a}, there is Z ∈ ℋ_{a}, where a_{Z} ∈ ℋ, such that Y ⊆ Z. Therefore, a_{Y} ⊂̃ a_{Z}. It follows that .
Proposition 3.9
Let (X, τ, A) be an STS, and let ℬ be a soft base of τ_{ω}. Then, (X, τ, A) is soft ω^{*}-paracompact if and only if for every ℋ ∈ SOC(X, τ_{ω}, A) with ℋ ⊆ ℬ, there is ℳ ∈ SOC(X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
Proof. Straightforward.
Corollary 3.10
Let (X, τ, A) be an STS. Then, (X, τ, A) is soft ω^{*}-paracompact if and only if for every ℋ ∈ SOC(X, τ_{ω}, A) with ℋ ⊆ {F − H : F ∈ τ and H ∈ CSS(X, A)}, there is ℳ ∈ SOC (X, τ_{ω}, A) such that ℳ is soft locally finite in (X, τ, A) and ℳ ⪯̃ ℋ.
Lemma 3.11
Let (X, τ, A) be an STS. If ℬ is a soft base for τ, then {B − H : B ∈ ℬ and H ∈ CSS(X, A)} is a soft base for τ_{ω}.
Proof. Straightforward.
Theorem 3.12
Let (X, τ, A) be an STS, where A = {a}. Then, (X, τ, A) is soft ω^{*}-paracompact if and only if the TS (X, τ_{a}) is ω^{*}-paracompact.
Proof. Necessity
We assume that (X, τ, A) is soft ω^{*}-paracompact. Let . By Proposition 2.9, we have (τ_{a})_{ω} = (τ_{ω})_{a}, and thus .
Then, {a_{U} : } ∈ SOC(X, τ_{ω}, A). As (X, τ, A) is soft ω^{*}-paracompact, there is ℋ ∈ SOC(X, τ_{ω}, A) such that ℋ is soft locally finite in (X, τ, A) and . Let .
Claim
(1) .
(2) is locally finite in (X, τ_{a}).
(3) .
Proof of Claim
1. As ℋ ⊆ τ_{ω}, we have and by Proposition 2.9, it follows that . In addition, as ℋ ∈ SOC(X, τ_{ω}, A), we have ∪̃{H : H ∈ ℋ} = 1_{A}, and thus (∪̃{H : H ∈ ℋ}) (a) = ∪{H (a) : H ∈ ℋ} = X. It follows that .
2. Let x ∈ X. Then, a_{x} ∈ SP(X, A). As ℋ is soft locally finite in (X, τ, A), there is F ∈ τ such that a_{x} ∊̃ F, and {H ∈ ℋ: F∩̃H ≠ 0_{A}} is finite. Then, x ∈ F (a) ∈ τ_{a}. Furthermore, if for H ∈ ℋ, we have F (a) ∩ H (a) ≠ ∅︀, then F ∩̃ H ≠ 0_{A}. It follows that {H ∈ ℋ: F (a) ∩ H (a) ≠ ∅︀} is finite. Therefore, is locally finite in (X, τ_{a}).
3. Let , where H ∈ ℋ. As , there is such that H ⊂̃ a_{U}; thus, H (a) ⊆ U. This shows that .
Sufficiency
We assume that (X, τ_{a}) is ω^{*}-paracompact. Let ℋ ∈ SOC(X, τ_{ω}, A). Then, {H (a) : H ∈ ℋ} ∈ OC(X, (τ_{ω})_{a}) = OC(X, (τ_{a})_{ω}). As (X, τ_{a}) is ω^{*}-paracompact, there is such that is locally finite in (X, τ_{a}) and . Let .
Claim
(1) ℳ ∈ SOC(X, τ_{ω}, A).
(2) ℳ is soft locally finite in (X, τ, A).
(3) ℳ ⪯̃ ℋ.
Proof of Claim
(1) As , we have ℳ ⊆ τ_{ω}. In addition, as , we have , and thus . It follows that ℳ ∈ SOC(X, τ_{ω}, A).
(2) Let a_{x} ∈ SP(X, A). As is locally finite in (X, τ_{a}), there is O ∈ τ_{a} such that x ∈ O and { : O ∩ V ≠ ∅︀} is finite. Then, a_{x} ∊̃ a_{O} ∈ τ. If a_{V} ∈ ℳ, where and a_{O} ∩̃ a_{V} ≠ 0_{A}, then (a_{O}∩̃a_{V}) (a) = O ∩V ≠ ∅︀. It follows that ℳ is soft locally finite in SOC(X, τ, A).
(3) Let a_{V} ∈ ℳ, where . As , there is H ∈ ℋ such that V ⊆ H (a), and thus a_{V} ⊂̃ H. This shows that ℳ ⪯̃ ℋ.
Theorem 3.13
Let X be an initial universe, and let A be a set of parameters. Moreover, let {ℑ_{a} : a ∈ A} be an indexed family of topologies on X. Then, $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft ω^{*}-paracompact if and only if (X, ℑ_{a}) is ω^{*}-paracompact for all a ∈ A.
Proof. Necessity
We assume that $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft ω^{*}-paracompact. Let b ∈ A. To show that (X, ℑ_{b}) is ω^{*}-paracompact, let . We set . Then, $\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$. By Proposition 2.10, $\underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega$, and thus $\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$. As $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ is soft ω^{*}-paracompact, there is $\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ such that ℳ is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$ and ℳ ⪯̃ ℋ. By Proposition 2.10, $\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$. Let ℳ_{b} = {M (b) : M ∈ ℳ}.
Claim
(1) ℳ_{b} ∈ OC(X, (ℑ_{b})_{ω}).
(2) ℳ_{b} is locally finite in (X, ℑ_{b}).
(3) .
Proof of Claim
1. As $\mathcal{M}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}$, by the definition of ℑ_{b}, we have ℳ_{b} ⊆ (ℑ_{b})_{ω}. As $\mathcal{M}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega},A\right)$, we have ∪̃{M : M ∈ ℳ} = 1_{A}; hence, ∪̃{M : M ∈ ℳ})(b) = ∪{M(b) : M ∈ ℳ} = X. Therefore, ℳ_{b} ∈ OC(X, (ℑ_{b})_{ω}).
2. Let x ∈ X. Then, b_{x} ∈ SP(X, A). As ℳ is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$, there is $F\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$ such that b_{x} ∊̃ F, and {M ∈ ℳ: F∩̃M ≠ 0_{A}} is finite. Thus, x ∈ F(b) ∈ ℑ_{b}. If M ∈ ℳ is such that F(b) ∩ M(b) ≠ ∅︀, then F ∩̃ M ≠ 0_{A}. It follows that ℳ_{b} is locally finite in (X, ℑ_{b}).
3. Let M(b) ∈ ℳ_{b} − {∅︀}, where M ∈ ℳ. As ℳ ⪯̃ ℋ, there is H ∈ ℋ such that M ⊂̃ H, and thus M(b) ⊆ H(b). As M(b) ≠ ∅︀, we have H(b) ≠ ∅︀, and thus there is such that H = b_{U} and H(b) = U. It follows that .
Sufficiency
We assume that (X, ℑ_{a}) is ω^{*}-paracompact for all a ∈ A. Let ℬ = {a_{Y} : a ∈ A and Y ∈ (ℑ_{a})_{ω}}. Then, ℬ is a soft base of $\underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega$. We apply Proposition 3.9. Let $\mathscr{H}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$ with ℋ ⊆ ℬ. For each a ∈ A, let ℋ_{a} = {Y ⊆ X : a_{Y} ∈ ℋ}. Then, for all a ∈ A, we have ℋ_{a} ∈ OC (X, (ℑ_{a})_{ω}_{,}), where (X, ℑ_{a}) is ω^{*}-paracompact. Thus, there is ℳ_{a} ∈ OC (X, (ℑ_{a})_{ω}) such that ℳ_{a} is locally finite in (X, ℑ_{a}) and ℳ_{a} ⪯ ℋ_{a}. Let .
Claim
(1) $\mathcal{G}\in SOC\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$.
(2) is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
(3) .
Proof of Claim
1. For all a ∈ A, we have ℳ_{a} ⊆ (ℑ_{a})_{ω}, and therefore $\mathcal{G}\subseteq {\displaystyle \underset{a\in A}{\oplus}}{({\Im}_{a})}_{\omega}={\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega}$. For all a ∈ A, we have ℳ_{a} ∈ OC (X, (ℑ_{a})_{ω}), and thus (∪̃{a_{Y} : a ∈ A and Y ∈ ℳ_{a}}) (a) = ∪{Y : Y ∈ ℳ_{a}} = X. Therefore, ∪̃{a_{Y} : a ∈ A and Y ∈ ℳ_{a}} = 1_{A}. It follows that $\mathcal{G}\in \mathcal{S}\mathcal{O}\mathcal{C}\hspace{0.17em}\hspace{0.17em}\left(X,{\left({\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}\right)}_{\omega},A\right)$.
2. Let b_{x} ∈ SP(X, A). As ℳ_{a} is locally finite in (X, ℑ_{b}), there is O ∈ ℑ_{b} such that x ∈ O, and {Y : O ∩ Y ≠ ∅︀} is finite. We have ${b}_{x}\in {b}_{O}\in {\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a}$. If b_{O} ∩̃ a_{Y} ≠ 0_{A}, then a = b and O ∩ Y ≠ ∅︀. This shows that {a_{Y} : a ∈ A and Y ∈ ℳ_{a}, and b_{O}∩̃a_{Y} ≠ 0_{A}} is finite. It follows that is soft locally finite in $\left(X,{\displaystyle \underset{a\in A}{\oplus}}{\Im}_{a},A\right)$.
3. Let with a ∈ A and Y ∈ ℳ_{a}. As ℳ_{a} ⪯ ℋ_{a}, there is Z ∈ ℋ_{a}, where a_{Z} ∈ ℋ, such that Y ⊆ Z. Therefore, a_{Y} ⊂̃ a_{Z}. It follows that .