so-metrizable spaces and images of metric spaces

Question 1.1

How characterize generalized metric spaces by images of metric spaces?

Question 1.2

How characterize so-metrizable spaces by images of metric spaces?

Definition 2.1

Let X be a space.

1. Let x ∈ P ⊂ X . P is called a sequential neighborhood of x in X [15] if whenever { x n } is a sequence in X converging to x , then { x n } is eventually in P .

2. Let P ⊂ X . P is called a sequentially-open subset of X [16] if P is a sequential neighborhood of x in X for each x ∈ P .

Remark 2.2

[17,18] The following are known.

1. P is a sequential neighborhood of x in X iff each sequence { x n } in X converging to x is frequently in P .

2. The intersection of finitely many sequential neighborhoods of x in X is a sequential neighborhood of x in X .

3. The intersection of finitely many sequentially-open subsets of X is a sequentially-open subset of X .

Definition 2.3

Let P = ⋃ { P x : x ∈ X } be a family of subsets of X , such that each A ∈ P x , x ∈ A .

1. P is called a network for X [19], if whenever x ∈ U with U open in X there is P ∈ P x such that x ∈ P ⊂ U . Here, the P x is called a network at x in X .

2. P is called an so-network for X [1], if each element of P is a sequentially-open subset of X , and the following (a) and (b) hold for each x ∈ X .

   1. P x is a network at x in X .

   2. If U , V ∈ P x , then W ⊂ U ⋂ V for some W ∈ P x .

   Here, the P x is called an so-network at x in X .

Definition 2.4

[1] A space X is called an so-metrizable space if X has a σ -locally finite so-network.

Remark 2.5

If X has a σ -locally finite network P consisting of sequentially-open subsets of X , then X is an so-metrizable space. In fact, put F = { ⋂ P ′ : P ′ is a finite subfamily of P } , then F is a σ -locally finite so-network for X .

Definition 2.6

Let P be a family of subsets of a space X .

1. Let K be a subset of X . P is called a cfp-cover of K [8], if P is a finite cover of K in X such that it can be precisely refined by some finite cover of K consisting of closed subsets of K .

2. Let K be a subset of X . P is called to have property c c for K [20], if whenever H is a compact subset of K , and H ⊂ U with U open in X , there is a subfamily P H of P such that P H is a cfp-cover of H and ⋃ P H ⊂ U .

3. P is called a cfp-network for X [21], if whenever K is a compact subset of X and K ⊂ U with U open in X , there is a finite subfamily P K of P such that P K is a cfp-cover of K and ⋃ P K ⊂ U .

4. P is called a strong k -network for X [20], if whenever K is a compact subset of X , there is a countable subfamily P K of P such that P K has property c c for K .

5. P is called a k -network for X [22], if whenever K ⊂ U with K compact in X and U open in X , there is a finite subfamily F of P such that K ⊂ ⋃ F ⊂ U .

Definition 2.7

Let f : X ⟶ Y be a mapping.

1. f is called an so-open mapping if whenever U is an open subset of X , f ( U ) is a sequentially-open subset of Y .

2. f is called a compact-covering mapping [23] if whenever K is a compact subset of Y , there is a compact subset C in X such that f ( C ) = K .

3. f is called a σ -mapping [8] if there is a base B for X such that f ( B ) is σ -locally-finite in Y .

Remark 2.8

1. Let f : X ⟶ Y be a mapping and B be a base for X . If f ( B ) is a sequentially-open subset of Y for each B ∈ B , then f is an so-open mapping.

2. If f : X ⟶ Y is a σ -mapping, then X is a metric space [24, Remark 1].

Lemma 2.9

[25] Let X be a compact space with a point-countable k -network. Then X is metrizable.

Lemma 2.10

[26] Let P be a point-countable cover of a space X . Then P is a cfp-network for X iff P is a strong k -network for X .

Lemma 2.11

Let P be a σ -locally finite so-network for a space X . Then P is a k -network for X .

Lemma 3.1

Let P be a σ -locally finite so-network for a space X . Then P is a strong k -network for X .

Proof

By Lemma 2.10, we only need to prove that P is a cfp-network for X . Let K be a compact subset of X and K ⊂ U with U open in X . Then it suffices to prove that there is a finite subfamily P K of P such that P K is a cfp-cover of K and ⋃ P K ⊂ U . By Lemma 2.11, P is a point-countable k -network for X . It follows that { P ⋂ K : P ∈ P } is a point-countable k -network for K . By Lemma 2.9, the subspace K is metrizable. Furthermore, for each x ∈ K , it is known that there is P x ∈ P x such that P x ⊂ U . In addition, U x is constructed as follows.

Let { V n } be a decreasing neighborhood base at x in K . For every P ∈ P , P ⋂ K is open in K because P ⋂ K is sequentially-open in K and K is metrizable. Put F x = P x ⋂ K . Then there is m ∈ N such that V m ⊂ F x ⊂ P x ⊂ U . Thus, F x is a neighborhood of x in K . So x ∈ int K ( F x ) ⊂ P x ⊂ U , where int K ( F x ) is the interior of F x in K . By regularity of K , there is an open (in K ) subset U x of K such that x ∈ U x ⊂ cl K ( U x ) = cl ( U x ) ⊂ int K ( F x ) , where cl K ( U x ) and cl ( U x ) are closures of U x in K and X , respectively. Consequently, U x is constructed.

Since { U x : x ∈ K } is an open (in K ) cover of K , there is a finite subset K ′ of K such that { U x : x ∈ K ′ } covers K . Put P K = { P x : x ∈ K ′ } . Then P K is a finite subfamily of P and ⋃ P K ⊂ U . Moreover, K = ⋃ { cl ( U x ) : x ∈ K ′ } and cl ( U x ) ⊂ int K ( F x ) ⊂ P x for each x ∈ K ′ . This shows that P K is a cfp-cover of K .□

Theorem 3.2

The following are equivalent for a space X .

1. X is an so-metrizable space.

2. X is an so-open, compact-covering, σ -image of a metric space.

3. X is an so-open, σ -image of a metric space.

Proof

(1) ⇒ (2): Let X be an so-metrizable space, and let P = { P β : β ∈ Λ } be a σ -locally finite so-network for X . For each n ∈ N , put Λ n = Λ and endow Λ n a discrete topology. Put

then M is a metric space and x b is unique for each b ∈ M . Define f : M ⟶ X by f ( b ) = x b , then f is continuous and onto, so f is a mapping. We only need to prove the following three facts.

Fact 1. f is a σ -mapping.

Put P ′ = { ⋂ F : F is a finite subfamily of P } , then P ′ is σ -locally finite because P is σ -locally finite. For each k ∈ N , Put B k = { ( ( ∏ n ≤ k { β n } ) × ( ∏ n > k Λ n ) ) ⋂ M : β n ∈ Λ n for each n ≤ k } and put B = ⋃ k ∈ N B k . Then B is a base for M . We only need to prove that f ( B ) ⊂ P ′ . Let B = ( ( ∏ n ≤ k { β n } ) × ( ∏ n > k Λ n ) ) ⋂ M ∈ B . It suffices to show that f ( B ) = ⋂ n ≤ k P β n . Let x = f ( c ) , where c = ( γ n ) ∈ B . Then γ n = β n for each n ≤ k , and hence x = f ( c ) ∈ P γ n = P β n for each n ≤ k . So f ( B ) ⊂ ⋂ n ≤ k P β n . Conversely, let x ∈ ⋂ n ≤ k P β n . Then there is a = ( α n ) ∈ M such that f ( a ) = x , that is, { P α n } is a network at x in X . Put γ n = β n for each n ≤ k and put γ n = α n − k for each n > k . It is clear that { P γ n } is a network at x in X . Put c = ( γ n ) , then x = f ( c ) and c ∈ B . So x ∈ f ( B ) , this proves that ⋂ n ≤ k P β n ⊂ f ( B ) . Thus, f ( B ) = ⋂ n ≤ k P β n .

Fact 2. f is a compact-covering mapping.

Let K be a non-empty compact subset of X . Since P is a strong k -network for X (by Lemma 3.1), there is a countable subfamily P K of P such that P K has property c c for K , and the family of cfp-covers of K consisting of elements of P K is countable. Let { P n : n ∈ N } be the family. For each n ∈ N , put P n = { P β : β ∈ Γ n } , where Γ n is a finite subset of Λ . Note that P n is a cfp-cover of K , there is a cover F n = { F β : β ∈ Γ n } of K consisting of compact subsets of K such that F β ⊂ P β for each β ∈ Γ n . Put C = { ( β n ) ∈ ∏ n ∈ N Γ n : ⋂ n ∈ N F β n ≠ ∅ } . It suffices to prove the following (a), (b) and (c).

1. C ⊂ M and f ( C ) ⊂ K .

   Let b = ( β n ) ∈ C , then ⋂ n ∈ N F β n ≠ ∅ . Choose x ∈ ⋂ n ∈ N F β n . We only need to prove that { P β n : n ∈ N } is a network at x in X , then b ∈ M and f ( b ) = x ∈ K , thus C ⊂ M and f ( C ) ⊂ K . Let V be a neighborhood of x in X . By regularity of K , there is an open neighborhood W of x in the subspace K such that W ⊂ E ⊂ V , where E is the closure of W in the subspace K . Since E is a compact subset of K and P K has c c -property on K , there is a finite family F ′ of P K such that F ′ is a cfp-cover of E and ⋃ F ′ ⊂ V . Similarly, since K − W is a compact subset of K and K − W ⊂ X − { x } , there is a finite family F ″ of P K such that F ″ is a cfp-cover of K − W and ⋃ F ″ ⊂ X − { x } . Put F = F ′ ⋃ F ″ , then F is a cfp-cover of K , so F = P n for some n ∈ N . Note that ⋃ F ″ ⊂ X − { x } and x ∈ F β n ⊂ P β n ∈ P n = F = F ′ ⋃ F ″ . So P β n ∉ F ″ . It follows that P β n ∈ F ′ , and hence P β n ⊂ V . Thus, we prove that { P β n : n ∈ N } is a network at x in X .

2. K ⊂ f ( C ) .

   Let x ∈ K . For each n ∈ N , choose β n ∈ Γ n such that x ∈ F β n . Put b = ( β n ) , then b ∈ C . By the same method as in the proof of (a), it is easy to prove that f ( b ) = x . So K ⊂ f ( C ) .

3. C is a compact subset of M .

   Note that ∏ n ∈ N Γ n is a compact subset of ∏ n ∈ N Λ n . It suffices to prove that C is a closed subset of ∏ n ∈ N Γ n . Obviously, C ⊂ ∏ n ∈ N Γ n . Put b = ( β n ) ∈ ∏ n ∈ N Γ n − C , then ⋂ n ∈ N F β n = ∅ . By the compactness of K , there is n 0 ∈ N such that ⋂ n ≤ n 0 F β n = ∅ . Let W = ( ∏ n ≤ n 0 { β n } ) × ( ∏ n > n 0 Γ n ) , then W is an open subset of ∏ n ∈ N Γ n such that b ∈ W and W ⋂ C = ∅ . Therefore, C is a closed subset of ∏ n ∈ N Γ n .

Fact 3. f is an so-open mapping.

Put B k = { ( ( ∏ n ≤ k { β n } ) × ( ∏ n > k Λ n ) ) ⋂ M : β n ∈ Λ n for each n ≤ k } and put B = ⋃ k ∈ N B k . Then B is a base for M . Whenever B ∈ B , then B = ( ( ∏ n ≤ k { β n } ) × ( ∏ n > k Λ n ) ) ⋂ M for some k ∈ N and β n ∈ Λ n ( n ≤ k ). In the proof of Fact 1, we obtain that f ( B ) = ⋂ n ≤ k P β n . Note that P β n is a sequentially-open subset of Y for each n ≤ k . By Remark 2.2(3), f ( B ) is a sequentially-open subset of Y . It follows that f is an so-open mapping from Remark 2.8(1).

(2) ⇒ (3): It is clear.

(3) ⇒ (1): Let f : M ⟶ X be an so-open, σ -mapping, where M is a metric space. Since f is a σ -mapping, there is a base B of M such that f ( B ) is σ -locally-finite in X . By Remark 2.5, it suffices to prove the following two facts.

Fact 1. f ( B ) is a network for X .

If x ∈ X and U is a neighborhood of x , then f − 1 ( x ) ⊂ f − 1 ( U ) . Choose b ∈ f − 1 ( x ) ⊂ f − 1 ( U ) . B is a base for M , so there is B ∈ B such that b ∈ B ⊂ f − 1 ( U ) . Thus, x = f ( b ) ∈ f ( B ) ⊂ f f − 1 ( U ) = U and f ( B ) ∈ f ( B ) . This proves that f ( B ) is a network for X .

Fact 2. Each element of f ( B ) is a sequentially-open subset of X .

Let B ∈ B , then B is an open subset of M . f is an so-open mapping, so f ( B ) is a sequentially-open subset of X . This proves that each element of f ( B ) is a sequentially-open subset of X .□

Definition 3.3

A network P = ⋃ { P x : x ∈ X } is called an s n -network for X [29] if the following conditions hold for each x ∈ X .

1. P x is a network at x in X .

2. If U , V ∈ P x , then W ⊂ U ⋂ V for some W ∈ P x .

3. Each element of P x is a sequential neighborhood of x in X .

Definition 3.4

A mapping f : X ⟶ Y is called an s n -open mapping [28] if for each y ∈ Y , there is an s n -network P y at y in Y such that for each x ∈ f − 1 ( y ) , whenever U is a neighborhood of x in X , then P ⊂ f ( U ) for some P ∈ P y .

Proposition 3.5

Let f : X ⟶ Y be a mapping. Then the following are equivalent.

1. f is an so-open mapping.

2. f is an s n -open mapping.

Proof

(1) ⇒ (2): Let f be an so-open mapping. For each x ∈ X , let B x be an open neighborhood base at x in X . For each y ∈ Y , put P y ′ = ⋃ { { f ( B ) : B ∈ B x } : x ∈ f − 1 ( y ) } and P y = { ⋂ P ′ : P ′ is a finite subfamily of P y ′ } . Now we prove that P y is an s n -network at y in Y . First, let y ∈ G with G open in Y . Choose x ∈ f − 1 ( y ) ⊂ f − 1 ( G ) . Then there is B ∈ B x such that x ∈ B ⊂ f − 1 ( G ) . Thus, y = f ( x ) ∈ f ( B ) ⊂ f f − 1 ( G ) = G and f ( B ) ∈ P y . So P y is a network at y in Y . Second, it is clear that P y is closed under finite intersections. Third, we need to prove that each element of P y is a sequential neighborhood of y in Y . By Remark 2.2(2), it suffices to prove that each element of P y ′ is a sequential neighborhood of y in Y . Let P ∈ P y ′ , then there is B ∈ B x such that P = f ( B ) for some x ∈ f − 1 ( y ) . Since f is an so-open mapping and B is an open subset of X , P = f ( B ) is a sequentially-open subset of Y . It follows that P is a sequential neighborhood of y in Y . By the above, P y is an s n -network at y in Y . If x ∈ f − 1 ( y ) and W is a neighborhood of x in X , then there is B ∈ B x such that x ∈ B ⊂ W . Put P = f ( B ) , then P ⊂ f ( W ) for some P ∈ P y . Thus, f is an s n -open mapping.

(2) ⇒ (1): Let f be an s n -open mapping. If y ∈ Y , then there is an s n -network P y at y in Y such that for each x ∈ f − 1 ( y ) , whenever U is a neighborhood of x in X , then P ⊂ f ( U ) for some P ∈ P y . Let W be an open subset of X and y ∈ f ( W ) . It suffices to prove that f ( W ) is a sequential neighborhood of y in Y . In fact, choose x ∈ W ⋂ f − 1 ( y ) . Since W is a neighborhood of x in X , there is P ∈ P y such that P ⊂ f ( W ) . Note that P is a sequential neighborhood of y in Y . It follows that f ( W ) is a sequential neighborhood of y in Y .□

Remark 3.6

s n -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable) that appeared in some relevant papers can be replaced with so-open mapping.