On nearly Hurewicz spaces

Definition 1.1

[6] A space X is said to be strongly star-Hurewicz (star-Hurewicz) if it satisfies the selection hypothesis SSH (resp., SH).

Definition 1.2

[14] A topological space X is a γ–set if for each sequence {𝓤_n : n ∈ N} of ω–covers of X there exists a sequence {𝓥_n : n ∈ N} such that for every n ∈ N, 𝓥_n ∈ 𝓤_n and {𝓥_n : n ∈ N} is a γ–cover of X.

Definition 1.3

A space X is called nearly-compact [17] if for every open cover 𝓤 of X has a finite subcollection 𝓥 such that ∪_V∈𝓥 int(cl(V)) = X.

Definition 1.4

A space X to be semi-Hurewicz [18] if for each sequence (𝓤_n : n ∈ N) of semi open covers of X there exists a sequence (𝓥_n : n ∈ N) such that for each n ∈ N, 𝓥_n is a finite subset of 𝓤_n and for each x ∈ X, x ∈ ∪ {V : V ∈ 𝓥_n} for all but finitely many n.

Definition 2.1

A space X is said to have nearly Hurewicz property if for each sequence (𝓤_n : n ∈ N) of open covers of X there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of 𝓤_n and for each x ∈ X, x ∈ ∪ {int(cl(V)) = scl(V) : V ∈ 𝓥_n} for all but finitely many n.

Remark 2.2

Example 2.3

(1) Real line with the usual Euclidean topology is Hurewicz so is nearly Hurewicz.

Example 2.4

Sorgenfrey line is not almost Menger (see Example 6d, [19]) it can not be almost Hurewicz and hence Sorgenfrey line is not nearly Hurewicz.

Lemma 2.5

[20] In a topological space X if O is open, then scl(O) = int(cl(O)).

Example 2.6

Let X be an uncountable set and p ∈ X. Then 𝓣_p = {O ⊆ X; p ∈ O or O = ϕ} is uncoutable particular point topology on X. Uncountable particular point topology is not Lindelöf [21] so it can not be Menger and can not be Hurewicz because every Menger space is Lindelöf and every Hurewicz space is Menger. To show that X is nearly Hurewicz we will show that for each x ∈ X, {O ∈ 𝓣 : x ∉ int(cl(O))} is a finite subcollection. As for A ⊆ X and A ∈ 𝓣_p, implies p ∈ A. Thus no closed set other than X contains p. Hence closure of any open set except than ϕ is X. This implies cl(A) = X. Therefore the collection of interior of closure of open sets is {ϕ, X} and x ∉ ϕ, for all x ∈ X. Hence X is nearly Hurewicz.

Example 2.7

Let R be the set of reals numbers, I the set of irrational numbers and Q the set of rational numbers and for each irrational x we choose a sequence {r_i : i ∈ N} of rational numbers converging to x in the Euclidean topology. The rational sequence topology τ is then defined by declaring both R and ϕ to be open, each rational open and selecting the sets U_α(x) = {x_α,i : i ∈ N} ∪ {x} as a basis for the irrational point x. If r ∈ Q, then the closure of {r} with respect to τ is equal {r}, and for every x ∈ I, the closure of U_α(x) is equal U_α(x). For every n ∈ N, 𝓤_n = {r : r ∈ Q} ∪ {U_n(x) : x ∈ I} is an open cover of (R, τ). (R, τ) does not have the almost (nearly) Hurewicz property because (R, τ) is not almost(nearly) Menger [22]. On the other hand, (R, τ) is weakly Hurewicz, because Q is dense in (R, τ) and each x ∈ Q belongs to cl(∪ 𝓥_n) for all n and scl(∪ 𝓥_n) = int(cl(∪ 𝓥_n)) = R.

Definition 2.8

A subset B of a topological space X is called s-regular open (resp. s-regular closed) if B = int(scl(B))(resp. B = cl(sInt(B))).

Theorem 2.9

A topological space X is nearly Hurewicz if and only if for each sequence (𝓤_n : n ∈ N) of covers of X by s-regular open sets, there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of 𝓤_n and each x ∈ X, x ∈ ∪ {V : V ∈ 𝓥_n} for all but finitely many n.

Proof

Let X be a nearly Hurewicz space. Let (𝓤_n : n ∈ N) be a sequence of covers of X by s-regular open sets. By assumption, there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of 𝓤_n and each x ∈ X, x ∈ ∪ {V = int(cl(V)) : V ∈ 𝓥_n} for all but finitely many n.

Conversely, let (𝓤_n : n ∈ N) be a sequence of open covers of X. Let ( Un′ : n ∈ N) be a sequence defined by Un′ = {Int(scl(U)) : U ∈ 𝓤_n}. Then each Un′ is a cover of X by s-regular open sets.

By hypothesis there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of Un′ and each x ∈ X, x ∈ ∪ {V : V ∈ 𝓥_n} for all but finitely many n. By construction, for each n ∈ N and V ∈ 𝓥_n there exists U_V ∈ 𝓤_n such that V = Int(scl(U_V)). Since Int(scl(U_V)) ⊆ scl(U_V) = Int(cl(U_V)) hence, each x ∈ X, x ∈ ∪ {int(cl(U_V)) : V ∈ 𝓥_n} for all but finitely many n. This implies that X is a nearly Hurewicz space.□

Theorem 2.10

For a topological space X the following statements are equivalent:

1. X is nearly Hurewicz;

2. X satisfies U_fin(𝓡𝓞, 𝓡𝓞),

where, 𝓡𝓞 denotes the collection of regular open sets.

Proof

(1) ⇒ (2) Let (𝓤_n : n ∈ N) be a sequence of regular open covers of X. Since X is nearly Hurewicz space there exists a sequence (𝓥_n : n ∈ N) such that for each n, 𝓥_n is a finite subset of 𝓤_n and for each x ∈ X x ∈ ⋃_n∈N ⋃ {Int(Cl(V) : V ∈ 𝓤_n} for all but finitely many n. Since Int(Cl(V)) = V for each n and each V ∈ 𝓥_n we conclude that (2) is satisfied.

(2) ⇒ (1) Let (𝓤_n : n ∈ N) be a sequence of open covers of X. Define for each n ∈ N, 𝓥_n = {Int(Cl(U)) : U ∈ 𝓤_n}. Then (𝓥_n : n ∈ N) is a sequence of regular open covers of X. By (2) there is a sequence (𝓦_n : n ∈ N) such that 𝓦_n is a finite subset of 𝓥_n for each n ∈ N, and each x ∈ X belongs to ⋃_n∈N ⋃ {W : W ∈ 𝓦_n} for all but finitely many n. Pick for each n and each W ∈ 𝓦_n a set U_W ∈ 𝓤_n with W = Int(Cl(U_W)) and set 𝓗_n = {U_W : W ∈ W_n}, a finite subset of 𝓤_n. Since ⋃_n∈N ⋃ {W : W ∈ 𝓦_n} = ⋃_n∈N ⋃ {Int(Cl(U_W)) : W ∈ 𝓦_n} we conclude that X is nearly Hurewicz.□

Definition 2.11

Let X and Y be topological spaces. A mapping f : X ⟶ Y is nearly continuous if for each s-regular open set B ⊂ Y, f^–1(B) is open in X. Every continuous mapping is nearly continuous.

Lemma 2.12

If f : X ⟶ Y is nearly continuous and open mapping, then for every s-regular open set B in Y, int(cl(f^–1(B))) ⊆ f^–1(int(cl(B))).

Lemma 2.13

If f : X ⟶ Y is nearly continuous and open mapping, then for every open set A in Y, f(int(cl((A)))) ⊆ int(cl(f(A))).

Theorem 2.14

Let f : X ⟶ Y be a nearly continuous open mapping from a nearlry Hurewicz space X onto Y. Then Y is nearly Hurewicz.

Proof

Let (𝓤_n : n ∈ N) be a sequence of covers of Y by s-regular open sets. Let Un′ = {f^–1(U) : U ∈ 𝓤_n} for each n ∈ N. Then ( Un′ : n ∈ N) is a sequence of open covers of X. By hypothesis, there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of Un′ and each x ∈ X, x ∈ ∪ {int(cl(V)) : V ∈ 𝓥_n} for all but finitely many n. For each n ∈ N and V ∈ 𝓥_n we can choose U_V ∈ 𝓤_n such that V = f^–1(U_V). Let 𝓦_n = {U_V : V ∈ 𝓥_n}. If y = f(x) ∈ Y, then there exists n ∈ N and V ∈ 𝓥_n such that x ∈ scl(V). Therefore x ∈ int(cl(f^–1(U_V))) ⊂ f^–1(int(cl(U_V))) = f^–1(scl(U_V)) = f^–1(U_V). Hence y = f(x) ∈ U_V ∈ 𝓦_n for all but finitely many n.□

Definition 2.15

Let X and Y be topological spaces. A mapping f : X ⟶ Y is nearly open if f^–1(int(cl(B))) ⊆ int(cl(f^–1(B))) for any subset B of Y.

Theorem 2.16

If f : X → Y is nearly open and perfect continuous mapping and Y is a nearly Hurewicz space, then X is a nearly Hurewicz space.

Proof

Let (𝓤_n : n ∈ N) be a sequence of open covers of X. Then due to perfect continuity of f, for each y ∈ Y and every n ∈ N, there is a finite subfamily 𝓤_ny of 𝓤_n such that f^–1(y) ⊂ ∪ 𝓤_ny. Let U_ny = ∪ 𝓤_ny. Then V_ny = Y – f(X ∖ U_ny) is an open neighborhood of y, since f is closed. For every n ∈ N, 𝓥_n = {V_ny : y ∈ Y}, is an open cover of Y. Y is nearly Hurewics so there exist a sequence ( Vn′ : n ∈ N) such that for every n ∈ N, Vn′ is a finite subset of 𝓥_n and for each y ∈ Y, y ∈ ∪ {int(cl(V)) : V ∈ Vn′ } for all but finitely many n. We may assume Vn′ = {V_nyi : i ≤ n^′} for each n ∈ N. For each n ∈ N, let Un′ = ∪_i≤n^′ 𝓤_nyi. Then Un′ is a finite subset of 𝓤_n. Since f is nearly open. Then

Hence X is nearly Hurewicz.□

Corollary 2.17

A continuous open surjective image of a nearly Hurewicz space is nearly Hurewicz.

Definition 2.18

Let X and Y be topological spaces. A mapping f : X ⟶ Y is sr–cotinuous if the inverse image of each open set V is s-regular open.

Theorem 2.19

An sr–continuous surjective image of a nearly Hurewicz space X is Hurewicz.

Proof

Let (𝓥_n : n ∈ N) be a sequence of open covers of Y. Since f is sr–continuous, for each n ∈ N and each V ∈ 𝓥_n the set f^–1(V) is s-regular open. Therefore, for each n, the set 𝓤_n = {f^–1(V) : V ∈ 𝓥_n} is a cover of X by s-regular open sets. As X is a nearly Hurewicz there exists a sequence (𝓖_n : n ∈ N) such that for each n, 𝓖_n is a finite subset of 𝓤_n and each x ∈ X, x ∈ ∪ {G : G ∈ 𝓖_n} for all but finitely many n. Then 𝓦_n = {f(G) : G ∈ 𝓖_n} is a finite subset of 𝓥_n for each n ∈ N and y ∈ 𝓦_n for n > n_o ∈ N. This means that Y is a Hurewicz space.□

Definition 2.20

A mapping f : X ⟶ Y is strongly s–θ continuous if for each x ∈ X and each open set V in Y containing f(x) there is an open set U in X containing x such that f(int(cl(U))) is a subset of V.

Theorem 2.21

Let f : X ⟶ Y be strongly s-θ continuous surjection and X be a nearly Hurewicz space, then Y is Hurewicz.

Proof

Let (𝓥_n : n ∈ N) be a sequence of open covers of Y. Let x ∈ X. For each n ∈ N there is a set V_x,n ∈ 𝓥_n containing f(x). Since f is strongly s-θ-continuous there is an open set U_x,n in X containing x such that f(int(cl(U_x,n))) is a subset of V_x,n. Therefore for each n, the set 𝓤_n = {U_x,n : x ∈ X} is an open cover of X. As X is nearly Hurewicz space so there exists a sequence (𝓕_n)_n∈N of finite sets such that for each n, 𝓕_n is a subset of 𝓤_n and each x ∈ X, x ∈ ∪{int(cl(F)) : F ∈ 𝓕_n} for n > n_o ∈ N. To each F ∈ 𝓕_n assign a set W_F ∈ 𝓥_n with f(int(cl(F))) a subset of W_F and put 𝓦_n = {W_F : F ∈ 𝓕_n}. We obtain the sequence (𝓦_n)_n∈N of finite subsets of 𝓥_n n ∈ N, which witnesses for (𝓥_n : n ∈ N) that Y is a Hurewicz space.□

Definition 2.22

[23] A mapping f : X ⟶ Y is called weakly continuous, if for each open set U in X containing x and there exists an open set V in Y containing f(x) such that f(U) ⊆ cl(V).

Definition 2.23

A mapping f : X ⟶ Y is called s-weakly continuous, if for each open set U in X containing x there exists an open set V in Y containing f(x) such that f(U) ⊆ int(cl(V)).

Theorem 2.24

Let f : X ⟶ Y be an s-weakly continuous surjection and X be a Hurewicz space the Y is nearly Hurewicz.

Proof

Let (𝓥_n : n ∈ N) be a sequence of open covers of Y. Let x ∈ X. For each n ∈ N and each open set U_x,n containing x there is a set V_x,n ∈ 𝓥_n containing f(x) such that f(U_x,n) ⊆ int(cl(V_x,n)). The set 𝓤_n = {U_x,n : x ∈ X} is an open cover of X. Apply the fact that X is a Hurewicz space to the sequence (𝓤_n)_n∈N and find a sequence (𝓕_n)_n∈N such that for each n, 𝓕_n is a finite subset of 𝓤_n and each x ∈ X, x ∈ ∪ (𝓕_n)_n>no∈N. To each n and each F ∈ 𝓕_n assign a set V_F ∈ 𝓥_n such that f(F) is a subset int(cl(V_F)) and put 𝓦_n = {V_F : F ∈ 𝓕_n}. Then

that is Y is a nearly Hurewicz space.□

Definition 2.25

An open cover 𝓤 of a space X is a nearly γ–cover if it is infinite and for every x ∈ X, {U ∈ 𝓤 : x ∉ int(cl(U))} is finite.

Definition 2.26

A topological space X is a nearly γ -set if for each sequence (𝓤_n : n ∈ N) of ω -covers of X there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n ∈ 𝓤_n and {𝓥_n : n ∈ N} is a nearly γ -cover of X.

Definition 2.27

A mapping f : X ⟶ Y is s-θ-continuous if for each x ∈ X, and each open set V in Y containing f(x) there is an open set U containing x such that f(int(cl(U))) ⊆ int(cl(V))

Theorem 2.28

Let f : X ⟶ Y be an s-θ -continuous surjection and let X be a nearly γ -set. Then Y is a nearly Hurewicz space.

Proof

Let (𝓥_n : n ∈ N) be a sequence of open covers of Y and x ∈ X. For each n ∈ N there is a set V_x,n ∈ 𝓥_n containing f(x). By assumption there is an open set U_x,n in X containing x and f(int(cl(U_x,n))) is a subset of int(cl(V_x,n)). For each n let 𝓤_n be the set of all finite unions of sets {U_x,n : x ∈ X}. Evidently each 𝓤_n is an ω -cover of X. X is a nearly γ -set so there exists a sequence (U_n)_n∈N such that for each n, U_n ∈ 𝓤_n and for each x ∈ X, the set {n ∈ N : x ∉ int(cl(U_n))} is finite.

Let U_n = U_x1,n ∪ U_x2,n ∪ … ∪ U_xi(n),n. To each U_xj,n, j ≤ i(n), assign a set V_xj,n ∈ 𝓥_n with f(int(cl(U_xj,n))) ⊂ int(cl(V_xj,n)). Let y = f(x) ∈ Y. Then from x ∈ int(cl(U_n)) for all n ≥ n_o for some n_o ∈ N, we get x ∈ int(cl(U_xp,n)) for some 1 ≤ p ≤ i(n) which implies y ∈ f(int(cl(U_xp,n))) ⊆ int(cl(V_xp,n)). If we put 𝓦_n = {V_xj,n : j = 1, 2, ..i(n)}, we obtain the sequence (𝓦_n; n ∈ N) of finite subsets of 𝓥_n, n ∈ N, such that each y ∈ Y belongs to all but finitely many sets of ∪{int(cl(W)) : W ∈ 𝓦_n}, that is Y is a nearly Hurewicz space. □

Definition 3.1

A topological space X is a nearly star-Hurewicz if for each sequence {𝓤_n : n ∈ N} of open covers of X there exists a sequence {𝓥_n : n ∈ N} such that for every n ∈ N, 𝓥_n is finite subset of 𝓤_n and each x ∈ X, x ∈ ∪{int(cl(St(∪ 𝓥_n, 𝓤_n))) : n ∈ N} for all but finitely many n.

Theorem 3.2

A topological space X is nearly star-Hurewicz if and only if for each sequence (𝓤_n : n ∈ N) of covers of X by s-regular open sets, there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of 𝓤_n and each x ∈ X, x ∈ ∪ {St(∪ 𝓥_n, 𝓤_n) : n ∈ N} for all but finitely many n.

Proof

Since every s-regular open set is open so direct part is obvious.

Converse: We will prove that X is a nearly star-Hurewicz space. Let (𝓤_n : n ∈ N) be a sequence of open covers of X. Let ( Un′ : n ∈ N) be a sequence defined by Un′ = {int(scl(U)) : U ∈ 𝓤_n}. Then each Un′ is a cover of X by s-regular open sets. By assumption, there exists a sequence (𝓥_n : n ∈ N) such that for every n ∈ N, 𝓥_n is a finite subset of Un′ and and each x ∈ X, x ∈ ∪ {int(cl(St(∪ 𝓥_n, Un′ ))) : n ∈ N} for all but finitely many n.

Claim 1: St(U, 𝓤_n) = St(int(scl(U)), 𝓤_n), for all U ∈ 𝓤_n.

Since U ⊂ int(scl(U)), it is obvious that St(U, 𝓤_n) ⊂ St(int(scl(U)), 𝓤_n). Now let x ∈ St(int(scl(U)), 𝓤_n) then there exists V ∈ 𝓤_n such that x ∈ V and V ∩ int(scl(U)) ≠ ∅. Then we have V ∩ U ≠ ∅ which implies x ∈ St(U, 𝓤_n) so St(Int(scl(U)), 𝓤_n) ⊂ St(U, 𝓤_n). Now for every V ∈ 𝓥_n we can choose U_V ∈ 𝓤_n such that V = Int(scl(U_v)), by construction. Let 𝓦_n = {U_V : V ∈ 𝓥_n}. We shall prove that x ∈ ∪ {int(cl(St(∪ 𝓦_n, 𝓤_n))) : n ∈ N} for all n ≥ n₀.

Let x ∈ X then there exists n ∈ N such that x ∈ int(cl(St(∪ 𝓥_n, Un′ ))). For every neighbourhood V of X, we have V ∩ St(∪ 𝓥_n, Un′ ) ≠ ϕ. Then there exist U ∈ 𝓤_n such that (V ∩ int(scl(U) ≠ ϕ) and (∪ 𝓥_n ∩ Int(scl(U)) ≠ ϕ) this implies (V ∩ U ≠ ϕ) and (∪ 𝓥_n ∩ U ≠ ∅). By claim 1 we have that ∪ 𝓦_n ∩ U ≠ ϕ, so x ∈ int(cl(St(∪ 𝓦_n, 𝓤_n))) for all but finitely many n. □

Theorem 3.3

The product X × Y of a nearly star-Hurewicz space X and a nearly compact space Y is nearly star Hurewicz.

Theorem 3.4

The nearly continuous open surjective image of a nearly star Hurewicz space is nearly star-Hurewicz.

Proof

Let f : X → Y be nearly continuous open surjection and X is nearly star Hurewicz. Let (𝓤_n : n ∈ N) be a sequence of covers of Y by s-regular open sets. Let Un′ = {f⁻¹(U) : U ∈ 𝓤_n} for each n ∈ N. Then ( Un′ : n ∈ N) is a sequence of open covers of X, by hypothesis, there exists a sequence ( Vn′ : n ∈ N) such that for every n ∈ N, Vn′ is a finite subset of Un′ and each x ∈ X, x ∈ ∪ {int(cl(St(∪ Vn′ , Un′ ))) : n ∈ N} for all but finitely many n.

Let 𝓥_n = {U : f⁻¹(U) ∈ Vn′ }. f⁻¹(∪ 𝓥_n) = ∪ Vn′ , and let x ∈ X, then there is n ∈ N such that x ∈ int(cl(St(f⁻¹(∪ 𝓥_n), Un′ )) for all but finitely many n. If y = f(x) ∈ Y, then y ∈ f(int(cl(St(f⁻¹(∪ 𝓥_n), Un′ ))) ⊆ int(cl(f(St(f⁻¹(∪ 𝓥_n), Un′ ))) ⊆ int(cl(St(∪𝓥_n, 𝓤_n))) for all but finitely many n. We will prove the last inclusion:

Suppose that f⁻¹(∪ 𝓥_n) ∩ f⁻¹(U) ≠ ϕ. Then also f(f⁻¹(∪ 𝓥_n)) ∩ f(f⁻¹(U)) ≠ ϕ, so ∪ 𝓥_n ∩ U ≠ ϕ.

So, the sequence (𝓥_n : n ∈ N) witnesses that X is a nearly star-Hurewicz. □

Theorem 3.5

If each finite power of a space X is nearly star-Hurewicz, then X satisfies Ufin∗(O,Ω_).

Proof

Let (𝓤_n : n ∈ N) be a sequence of open covers of X and consider N = N₁ ∪ N₂ ∪ … be a partition of N into infinitely many pairwise disjoint sets. for every k ∈ N and every j ∈ N_k. Let 𝓦_j = {U₁ × U₂ × … × U_k : U₁, U₂, …, U_k ∈ 𝓤_j} = Ujk . Then (𝓦_j : j ∈ N_k) is a sequence of open covers of X^k. Since X^k is nearly star-Hurewicz so we can choose a sequence (𝓗_j : j ∈ N_k) such that for each j, 𝓗_j is finite subset of 𝓦_j and each x ∈ X^k, x ∈ ∪ {int(cl(St(∪ H_j, 𝓦_j))) : j ∈ N_k} for all but finitely many j. For every j ∈ N_k and every H ∈ 𝓗_j we have H = U₁(H) × U₂(H) × … × U_k(H), where U_i(H) ∈ 𝓤_j for every i ≤ k. Now consider 𝓥_j = {U_i(H) : i ≤ k, H ∈ 𝓗_j}. Then for each j ∈ N_k, 𝓥_j is finite subset of 𝓤_j.

We claim that {int(cl(St(∪ 𝓥_n, 𝓤_n))) : n ∈ N} is an ω -cover of X. Let F = {x₁, …, x_p} be a finite subset of X. Then y = (x₁, …, x_p) ∈ X^p so that there is an n ∈ N_p such that y ∈ {int(cl(St(∪ H, 𝓤_n))); H ∈ 𝓗_n}. But H = U₁(H) × U₂(H) × … × U_p(H), where U₁(H), U₂(H), …, U_p(H) ∈ 𝓥_n. The point y belongs to some W ∈ 𝓦_n of the form V₁ × V₂ × … × V_p, 𝓥_i ∈ 𝓤_n for each i ≤ p, which meets U₁(H) × U₂(H) × … × U_p(H). This implies that for each i ≤ p, we have x_i ∈ int(cl(St(U_i(H), 𝓤_n))) ⊂ int(cl(St(∪ 𝓥_n, 𝓤_n))) for all but finitely many n, that is, F ⊂ int(cl(St(∪𝓥_n, 𝓤_n))). Hence X satisfy Ufin∗(O,Ω_). □

Definition 3.6

A space X is nearly strongly star-Hurewicz if for each sequence (𝓤_n : n ∈ N) of open covers of X there is a sequence (F_n : n ∈ N) of finite subsets of X such that each x ∈ X, x ∈ ∪ {int(cl(St(F_n, 𝓤_n))) : n ∈ N} for all but finitely many n.

Definition 3.7

[24] A space X is meta compact if every open cover 𝓤 of X has a point-finite open refinement 𝓥 (that is, every point of X belongs to at most finitely many members of 𝓥).

Theorem 3.8

Every nearly strongly star-Hurewicz meta compact space is a Hurewicz space.

Proof

Let X be nearly strongly star Hurewicz meta compact space. Let (𝓤_n : n ∈ N) be a sequence of open covers of X. For each n ∈ N, let 𝓥_n be a point-finite open refinement of 𝓤_n. Since X is nearly strongly star-Hurewicz, there is a sequence (F_n : n ∈ N) of finite subsets of X such that each x ∈ ∪ int(cl(St(F_n, 𝓥_n))) for all but finitely many n.

As 𝓥_n is point-finite open refinement and each F_n is finite, elements of each F_n belongs to finitely many members of 𝓥_n say V_n1, V_n2, V_n3, … V_nk. Let Vn′ = {V_n1, V_n2, V_n3, …, V_nk}. Then int(cl(St(F_n, 𝓥_n))) = ∪ Vn′ for each n ∈ N so we have that each x belongs to X belongs to ∪ Vn′ for all but finitely many n. For every V ∈ Vn′ choose U_V ∈ 𝓤_n such that V ⊂ U_V then for every n, 𝓦_n} = {U_V : V ∈ Vn′ } is a finite subfamily of 𝓤_n and each x belongs to X belongs to ∪ 𝓦_n for all but finitely many n, that is X is a Hurewicz space. □

Definition 3.9

[25] A space X is said to be meta Lindelöf if every open cover 𝓤 of X has a point-countable open refinement 𝓥.

Theorem 3.10

Every nearly strongly star Hurewicz meta Lindelöf space is a Lindelöf space.

Proof

Let X be nearly strongly star Hurewicz meta Lindelöf space. Let 𝓤 be an open covers of X and 𝓥 a point-countable open refinement of 𝓤 by hypothesis, there is a sequence (F_n : n ∈ N) of finite subsets of X such that each x belongs to x belongs to ∪ int(cl(St(F_n, 𝓥_n))) for all but finitely many n.

For every n ∈ N, denote by 𝓦_n the collection of all members of 𝓥 which intersect F_n. Since 𝓥 is point-countable and F_n is finite so 𝓦_n is countable. Therefor the collection 𝓦 = ∪ _n∈N 𝓦_n is a countable subfamily of 𝓥 and is a cover of X. For every W ∈ 𝓦 pick a member U_W ∈ 𝓤 such that W ∈ U_W. Then {U_W : W ∈ 𝓦} is a countable subcover of 𝓤. Hence X is a Lindelöf space. □

Theorem 3.11

A continuous image of a nearly strongly star-Hurewicz space is nearly strongly star-Hurewicz space.

Proof

Let f : X → Y be a continuous mapping from a nearly strongly star-Hurewicz space X onto a space Y. Let (𝓤_n : n ∈ N) be a sequence of open covers of Y. For each n ∈ N, let 𝓥_n = {f⁻¹(U) : U ∈ 𝓤_n}. Then (𝓥_n : n ∈ N) is a sequence of open covers of X. Since X is nearly strongly star-Hurewicz space, there exists a sequence (A_n : n ∈ N) of finite subsets of X such that for each x ∈ X, x ∈ ∪ int(cl(St(A_n, 𝓤_n))) for all but finitely many n. Thus (f(A_n) : n ∈ N) is a sequence of finite subsets of Y such that for each y ∈ Y, y ∈ int(cl(St(f(A_n), 𝓤_n))) for all but finitely many n. In fact, let y ∈ Y. Then there is x ∈ X such that f(x) = y. Hence x ∈ ∪ int(cl(St(A_n, 𝓥_n))) for all but finitely many n, which shows that Y is nearly strongly star-Hurewicz space. □

Theorem 3.11

If each finite power of a space X is nearly strongly star-Hurewicz space, then X satisfies SSfin∗ (O, Ω).

Proof

Let (𝓤_n : n ∈ N) be a sequence of open covers of X and consider N = N₁ ∪ N₂ ∪ … be a partition of N into infinite pairwise disjoint sets. For every k ∈ N and every j ∈ N_k. Let 𝓦_j = {U₁ × U₂ × … × U_k : U₁, U₂, …, U_k ∈ 𝓤_j} = Ujk . Then (𝓦_j : j ∈ N_k) is a sequence of open covers of X^k. Since X^k is nearly strongly star-Hurewicz space so we can choose a sequence (𝓥_j : j ∈ N_k) such that for each j, 𝓥_j is finite subset of X and each x ∈ X^k, x ∈ ∪ {int(cl(St(V, 𝓦_j))) : V ∈ 𝓥_j} for all but finitely many j. For every j ∈ N_k consider A_j a finite suset of X such that V_j ⊂ Atk .

We show that {int(cl(St(A_n, 𝓤_n))) : n ∈ N} is an ω -cover of X. Let F = {x₁, …, x_p} be a finite subset of X. Then (x₁, …, x_p) ∈ X^p so that there is an n ∈ N_p such that (x₁, …, x_p) ∈ int(cl(St(V_n, 𝓦_n))) ⊂ int(cl(St( Anp , 𝓦_n))) that is, F ⊂ int(cl(St(A_n, 𝓤_n))). □