PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 213–221 https://doi.org/10.2298/PIM1715213L

MAPPING i_{2} ON THE FREE PARATOPOLOGICAL GROUPS

Fucai Lin and Chuan Liu

Abstract. Let F P (X) be the free paratopological group over a topological space X. For each nonnegative integer n ∈ ℕ, denote by F P_{n} (X) the subset of F P (X) consisting of all words of reduced length at most n, and i_{n} by the natural mapping from (X ⊕ X^{−1} ⊕ {e})^{n} to F P_{n} (X). We prove that the natural mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → F P_{2}(X) is a closed mapping if and only if every neighborhood U of the diagonal Δ_{1} in X_{d} × X is a member of the finest quasi-uniformity on X, where X is a T_{1}-space and X_{d} denotes X when equipped with the discrete topology in place of its given topology.

1. Introduction

In 1941, free topological groups were introduced by Markov in [9] with the clear idea of extending the well-known construction of a free group from group theory to topological groups. Now, free topological groups have become a powerful tool of study in the theory of topological groups and serve as a source of various examples and as an instrument for proving new theorems, see [1].

As in free topological groups, Romaguera, Sanchis and Tkachenko in [12] defined free paratopological groups and proved the existence of the free paratopological group F P (X) for every topological space X. Recently, Elfard, Lin, Nickolas, and Pyrch have investigated some properties of free paratopological groups, see [2, 3, 7, 8, 10, 11].

For each nonnegative integer n ∈ ℕ, denote by F P_{n}(X) the subset of F P (X) consisting of all words of reduced length at most n, and i_{n} by the natural mapping from (X ⊕X^{−1} ⊕{e})^{n} to F P_{n}(X). Here we mainly improve some results of Elfard and Nickolas. The main result is that the natural mapping i_{2} : (X ⊕X_{d}^{−1} ⊕{e})^{2} → F P_{2}(X) is a closed mapping if and only if every neighborhood U of the diagonal Δ_{1} in X_{d} × X is a member of the finest quasi-uniformity on X, where X is a T_{1}-space and X_{d} denotes X when equipped with the discrete topology in place of its given topology.

2010 Mathematics Subject Classification: Primary 22A30; Secondary 54D10; 54E99; 54H99.

Key words and phrases: free paratopological groups; quotient mappings; closed mappings; finest quasi-uniformity.

Communicated by Rade Živaljević.

2. Preliminaries

All mappings are continuous. We denote by ℕ and ℤ the sets of all natural numbers and the integers, respectively. The letter e denotes the neutral element of a group. Readers may consult [1,4–6] for notations and terminology not explicitly given here.

Recall that a topological group G is a group G with a (Hausdorff) topology such that the product mapping of G × G into G is jointly continuous and the inverse mapping of G onto itself associating x^{−1} with an arbitrary x ∈ G is continuous. A paratopological group G is a group G with a topology such that the product mapping of G × G into G is jointly continuous.

Definition 2.1. [12] Let X be a subspace of a paratopological group G. Assume that

(1) The set X generates G algebraically, that is < ⟨X⟩ = G;

(2) Each continuous mapping f: X → H to a paratopological group H ex-

tends to a continuous homomorphism f̂: G → H.

Then G is called the Markov free paratopological group on X and is denoted by F P (X).

Again, if all the groups in the above definitions are Abelian, then we get the definition of the Markov free Abelian paratopological group on X which will be denoted by AP (X).

By [12], F P X and AP (X) exist for every space X and the underlying abstract groups of F P X and AP (X) are the free groups on the underlying set of the topological space X respectively. We denote these abstract groups by F P_{a}(X) and AP_{a}(X) respectively.

Since X generates the free group F P_{a}(X), each element g ∈ F P_{a}(X) has the form g = x_{1}^{ε_{1}} …x_{n}^{ε_{n}} , where x_{1}, …, x_{n} ∈ X and ε_{1}, …, ε_{n} = ±1. This word for g is called reduced if it contains no pair of consecutive symbols of the form xx^{−1} or x^{−1} x. It follows that if the word g is reduced and nonempty, then it is different from the neutral element of F P_{a}(X). For every nonnegative integer n, denote by F P_{n}(X) and AP_{n}(X) the subspace of paratopological groups F P (X) and AP (X) that consists of all words of reduced length ⩽ n with respect to the free basis X, respectively.

Let X be a T_{1}-space. For each n ∈ ℕ, denote by i_{n} the multiplication mapping from (X ⊕ X_{d}^{−1} ⊕ {e})^{n} to B_{n}(X), i_{n}(y_{1}, …, y_{n}) = y_{1} ⋯ y_{n} for every point (y_{1}, …, y_{n}) ∈ (X ⊕ X_{d}^{−1} ⊕ {e})^{n}, where X_{d}^{−1} denotes the set X^{−1} equipped with the discrete topology and B_{n}(X) denotes F P_{n}(X) or AP_{n}(X).

By a quasi-uniform space (X, 𝒰) we mean the natural analog of a uniform space obtained by dropping the symmetry axiom. For each quasi-uniformity 𝒰 the filter 𝒰^{−1} consisting of the inverse relations U^{−1} = {(y, x) : (x, y) ∈ U} where U ∈ 𝒰 is called the conjugate quasi-uniformity of 𝒰.

THE MAPPING i_{2} ON THE FREE PARATOPOLOGICAL GROUPS 215

Let X be a topological space. Then X_{d} denotes X when equipped with the discrete topology in place of its given topology. We denote the diagonals of X_{d} × X and X × X_{d} by Δ_{1} and Δ_{2}, respectively. In [10], the authors proved that X^{−1} is discrete in free paratopological group F P (X) and AP (X) over X if X is a T_{1}-space. We denote the sets {(x^{−1} , y) : (x, y) ∈ X × X} and {(x, y^{−1} ) : (x, y) ∈ X × X} by Δ_{1}^{∗} and Δ_{2}^{∗}, respectively.

3. Main results

First, we recall some results in the free paratopological groups.

Theorem 3.1. [3] If X is a T_{1}-space, then the mapping

i_{2}|_{i_{2}^{−1} (F P_{2} (X)∖F P_{1} (X))} : i_{2}^{−1} (F P_{2}(X) ∖ F P_{1}(X)) → F P_{2}(X) ∖ F P_{1}(X)

is a homeomorphism.

Theorem 3.2. [2] Let X be a T_{1}-space and let w = x_{1}^{ϵ_{1}}x_{2}^{ϵ_{2}} …x_{n}^{ϵ_{n}} be a reduced word in F P_{n}(X), where x_{i} ∈ X and ϵ_{i} = ±1, for all i = 1, 2, …, n, and if x_{i} = x_{i+1} for some i = 1, 2, …, n − 1, then ϵ_{i} = ϵ_{i+1} . Then the collection ℬ of all sets of the form U_{1}^{ϵ_{1}}U_{2}^{ϵ_{2}} …U_{n}^{ϵ_{n}}, where, for all i = 1, 2, …, n, the set U_{i} is a neighborhood of x_{i} in X when ϵ_{i} = 1 and U_{i} = {x_{i}} when ϵ_{i} = −1 is a base for the neighborhood system at w in F P_{n}(X).

Theorem 3.3. [2] Let X be a T_{1}-space and let w = ϵ_{1}x_{1} + ϵ_{2}x_{2} + ⋯+ ϵ_{n}x_{n} be a reduced word in AP_{n}(X), where x_{i} ∈ X and ϵ_{i} = ±1, for all i = 1, 2, …, n, and if x_{i} = x_{j} for some i, j = 1, 2, …, n, then ϵ_{i} = ϵ_{j}. Then the collection ℬ of all sets of the form ϵ_{1}U_{1} + ϵ_{2}U_{2} + ⋯+ ϵ_{n}U_{n}, where, for all i = 1, 2, …, n, the set U_{i} is a neighborhood of x_{i} in X when ϵ_{i} = 1 and U_{i} = {x_{i}} when ϵ_{i} = −1 is a base for the neighborhood system at w in AP_{n}(X).

Theorem 3.4. If X is a T_{1}-space, then the mapping

f = i_{2} |_{i_{2}^{−1} (AP_{2} (X)∖AP_{1} (X))} : i_{2}^{−1} (AP_{2}(X) ∖ AP_{1}(X)) → AP_{2}(X) ∖ AP_{1}(X)

is a 2 to 1, open and perfect mapping.

Proof. Obviously, f is a 2 to 1 mapping. Next, we shall prove that f is open and closed. Let C_{2}(X) = AP_{2}(X)∖AP_{1}(X) and C_{2}^{∗}(X) = i_{2}^{−1} (AP_{2}(X)∖AP_{1}(X)). Obviously, we have

C_{2}^{∗}(X) = (X × X) ⊕ (X_{d}^{−1} × X_{d}^{−1} ) ⊕ (X_{d}^{−1} × X) ∖ Δ_{1}^{∗} ⊕ (X × X_{d}^{−1} ) ∖ Δ_{2}^{∗}.

(1) The mapping f is open. Let (x_{1}^{ϵ_{1}}, x_{2}^{ϵ_{2}}) ∈ C_{2}^{∗}(X), where x_{1}, x_{2} ∈ X and x_{1} ≠ x_{2} if ϵ_{1} ≠ ϵ_{2}. Let U be a neighborhood of (x_{1}^{ϵ_{1}}, x_{2}^{ϵ_{2}}) in C_{2}^{∗}(X). By Theorem 3.3, f(U) is a neighborhood of x_{1}^{ϵ_{1}}x_{2}^{ϵ_{2}} in C_{2}(X). (Indeed, the argument is similar to the proof of [3, Theorem 3.4].) Therefore, f is open.

(2) The mapping f is closed. Let E be a closed subset of C_{2}^{∗}(X). To show that i_{2}(E) is closed in C_{2}(X) take w ∈ \overline{i_{2}(E)}. Next, we shall show that w ∈ i_{2}(E). Indeed, it is obvious that w has a reduced form w = ϵ_{1}x_{1} + ϵ_{2}x_{2}, where ϵ_{i} = 1 or -1 (i = 1, 2), x_{1}, x_{2} ∈ X and x_{1} ≠ x_{2} if ϵ_{1} ≠ ϵ_{2}.

Suppose that w = x + y ∉ i_{2}(E), where x = ϵ_{1}x_{1} and y = ϵ_{2}x_{2}. Then {(x, y), (y, x)} ∩ E = ∅. Since E is closed, we can pick open neighborhoods V (x) of x in X ∪ X_{d}^{−1} , V (y) of y in X ∪ X_{d}^{−1} such that (V (x) × V (y)) ∩ E = ∅ and (V (y) × V (x)) ∩ E = ∅. Let U = (V (x) × V (y)) ∪ (V (y) × V (x)). Then U is open. Since f is an open map, we have f(U) is a neighborhood of w and f(U)∩i_{2}(E) = ∅. This contradicts with w ∈ \overline{i_{2}(E)}. □

For an arbitrary space X, the mapping f: X → ℤ defined by setting f(x) = 1 for all x ∈ X is continuous, and thus extends to a continuous homomorphism f̂: AP (X) → ℤ. Therefore, the collection of sets Z_{n}(X) = f̂^{−1} ({n}) for n ∈ ℤ forms a partition of AP (X) into clopen subspaces.

For a T_{1}-space, define

g: (X_{d} × X) ⊕ (X × X_{d}) ⊕ ({e} × {e}) → AP_{2}(X) ∩ Z_{0}(X)

by

g(x, y) = \{ \begin{matrix}−x + y, & if (x, y) ∈ X_{d} × X; \\ x − y, & if (x, y) ∈ X × X_{d}; \\ e, & if x = y. \end{matrix}

Let g_{j} = i_{2} |_{i_{2}^{−1} (AP_{2} (X)∩Z_{j} (X))} for j = −2, …, 2, where

i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → AP_{2}(X).

Obviously, i_{2} = ⨁ _{j=−2}^{j=2} {g_{j} }, and i_{2} is a closed (resp. quotient) mapping if and only if each g_{j} is a closed (resp. quotient) mapping, where j = −2, …, 2. By Theorem 3.4, it is easy to see that g_{−2} and g_{2} are open and closed. Moreover, since −X occurs with the discrete topology and X occurs with its original topology in AP (X), the mappings g_{−1} and g_{1} are open and closed. Obviously, g is a closed (resp. quotient) mapping if and only if g_{0} is a closed (resp. quotient) mapping. Therefore, we have the following result:

Lemma 3.1. Let X be a T_{1}-space. Then i_{2} is a closed (resp. quotient) mapping if and only if g is a closed (resp. quotient) mapping.

Lemma 3.2. [3] Let X be a space and let Δ_{1} be the diagonal in the space X_{d} × X. Then Δ_{1} is closed if and only if X is T_{1}. Similarly for the diagonal Δ_{2} in the space X × X_{d}.

Suppose that 𝒰^{∗} is the finest quasi-uniformity of a space X. We say that P = {U_{i}}_{i∈ℕ} is a sequence of 𝒰^{∗} if each U_{i} ∈ 𝒰^{∗}. Put

ω 𝒰^{∗} = {P : P is a sequence of 𝒰^{∗}}.
For each n ∈ ℕ and P = {U_{i}}_{i∈ℕ} ∈ ^{ω} 𝒰^{∗}, let 𝒬_{n}(ℕ) = {A ⊂ ℕ : |A| = n},
W_{n}(P) = {−x_{1} + y_{1} − ⋯− x_{n} + y_{n} : (x_{j} , y_{j}) ∈ U_{i_{j}}

for j = 1, 2, …, n, {i_{1}, i_{2}, …, i_{n}} ∈ 𝒬_{n}(ℕ)}, and 𝒲_{n} = {W_{n}(P) : P ∈ ^{ω} 𝒰^{∗}}.

Remark 3.1. In the above definition, for P = {U_{i}}_{i∈ℕ} ∈ ^{ω} 𝒰^{∗}, there may exist i ≠ j such that U_{i} = U_{j}. In particular, for every U ∈ 𝒰^{∗}, we have {U_{i} = U}_{i∈ℕ} is also in ^{ω} 𝒰^{∗}. Moreover, the reader should note that the representation of elements of W_{n}(P) need not be a reduced representation.

Theorem 3.5. [7] For each n ∈ ℕ, the family 𝒲_{n} is a neighborhood base of e in AP_{2n} (X).

The proof of the following Theorem is a modification of [3, Theorem 3.10].

Theorem 3.6. Let X be a T_{1}-space. Then the mapping

i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → AP_{2}(X)

is a quotient mapping if and only if every neighborhood U of the diagonal Δ_{1} in X_{d} × X is a member of the finest quasi-uniformity 𝒰^{∗} on X.

Proof. Put Z = (X_{d} × X) ⊕ (X × X_{d}) ⊕ ({e} × {e}).

Necessity. Suppose that i_{2} is a quotient mapping. It follows from Lemma 3.1 that g: Z → AP_{2}(X) ∩ Z_{0}(X) is a quotient mapping. Let U be a neighborhood of Δ_{1} in X_{d} × X. Obviously, U ∪ (−U) is a neighborhood of Δ_{1} ∪ Δ_{2} in Z. Let P = {U_{n}}_{n∈ℕ} , where U_{n} = U for each n ∈ ℕ. Let W_{1}(P) = {−x + y : (x, y) ∈ U}. Then g^{−1} (W_{1}(P)) = U ∪(−U)∪{(e, e)} that is a neighborhood of Δ_{1}∪Δ_{2}∪{(e, e)} in Z, then W_{1}(P) is a neighborhood of e in AP_{2}(X)∩Z_{0}(X), and hence in AP_{2}(X). By Theorem 3.5, there exists Q ∈ ^{ω} 𝒰^{∗} such that W_{1}(Q) ⊂ W_{1}(P), where Q = {V_{n}}_{n∈ℕ} . Then V_{1} ⊂ U, hence U ∈ 𝒰^{∗}.

Sufficiency. Suppose that every neighborhood U of the diagonal Δ_{1} in X_{d} × X is a member of the finest quasi-uniformity 𝒰^{∗} on X. To show that i_{2} is a quotient mapping, it follows from Lemma 3.1 that it suffices to show that the mapping g: Z → AP_{2}(X) ∩ Z_{0}(X) is a quotient mapping. Take a subset A ⊂ AP_{2}(X) ∩ Z_{0}(X) such that g^{−1} (A) is open in Z. Put U = g^{−1} (A) ∩ (X_{d} × X) and V = g^{−1} (A) ∩ (X × X_{d}). Firstly, we show the following claim:

Claim: If e ∉ A, then A is open in AP_{2}(X) ∩ Z_{0}(X). Indeed, since e ∉ A, U ∩ Δ_{1} = ∅ and V ∩ Δ_{2} = ∅. By Lemma 3.2, Δ_{1} and Δ_{2} are closed in X_{d} × X and X × X_{d}, respectively, and X_{d} × X ∖ Δ_{1} and X × X_{d} ∖ Δ_{2} are open in X_{d} × X and X × X_{d}, respectively. Hence U ∪ V is open in the space i_{2}^{−1} (AP_{2}(X) ∖ AP_{1}(X)), and by Theorem 3.4, g(U ∪ V) = A is open in AP_{2}(X) ∩ Z_{0}(X).

Next we shall show that A is open in AP_{2}(X) ∩ Z_{0}(X). Take arbitrary a ∈ A. Then it suffices to show that A is an open neighborhood of a.

Case 1: a = e. Obviously, U and V are open neighborhoods of Δ_{1} and Δ_{2} in X_{d} × X and X × X_{d}, respectively. Therefore, S = U ∩ (V^{−1} ) is an open neighborhood of Δ_{1} in X_{d}×X, and thus S ∈ 𝒰^{∗}. Let W_{1}(R) = {−x+y : (x, y) ∈ S}, where R = {S_{n}}_{n∈ℕ} and S_{n} = S for each n ∈ ℕ. By Theorem 3.5, W_{1}(R) is a neighborhood of e in AP_{2}(X). Since S = U ∩ (V^{−1} ) and the definition of g, it is easy to see that W_{1}(R) ⊂ A. Therefore, A is a neighborhood of e in AP_{2}(X), hence in AP_{2}(X) ∩ Z_{0}(X).

Case 2: a ≠ e. Let W be an open neighborhood of a in AP_{2}(X) ∩ Z_{0}(X) such that e ∉ W. Then the set g^{−1} (A ∩ W) is open in Z, and it follows from the claim that A ∩ W is an open neighborhood of a in AP_{2}(X) ∩ Z_{0}(X). Hence A is open in AP_{2}(X) ∩ Z_{0}(X). □

The following theorem is the main result in [3], and some related concepts can be seen in [5]. Next, we shall improve this result in Theorem 3.9.

Theorem 3.7. [3] Let X be a T_{1}-space. Then the following statements are equivalent:

(1) The mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → F P_{2}(X) is a quotient mapping;

(2) Every neighborhood U of the diagonal Δ_{1} in X_{d} × X is a member of the finest quasi-uniformity 𝒰^{∗} on X;

(3) Every neighbornet of X is normal;

(4) The finest quasi-uniformity 𝒰^{∗} on X consists of all neighborhoods of the diagonal Δ_{1} in X_{d} × X;

(5) If N_{x} is a neighborhood of x for all x ∈ X, then there exists a neighborhood M_{x} of x such that ⋃ _{y∈M_{x}} M_{y} ⊂ N_{x} for all x ∈ X;

(6) If N_{x} is a neighborhood of x for all x ∈ X, then there exists a quasipseudometric d on X such that d_{x} is upper semi-continuous and B_{d}(x, 1) ⊂ N_{x} for all x ∈ X.

Let X be a set. Define j_{2}, k_{2} : X × X → F_{a}(X) by j_{2}(x, y) = x^{−1} y and k_{2}(x, y) = yx^{−1} .

Theorem 3.8. [3] Let X be a topological space. Then the collection ℬ of sets j_{2}(U) ∪ k_{2}(U) for U ∈ 𝒰^{∗} is a base of neighborhoods at the identity e in F P_{2}(X).

Now we can prove the main theorem in this paper.

Theorem 3.9. Let X be a T_{1}-space. Then the following statements are equivalent:

(1) The mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → F P_{2}(X) is a quotient mapping;

(2) The mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → AP_{2}(X) is a quotient mapping;

(3) The mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → F P_{2}(X) is a closed mapping;

(4) The mapping i_{2} : (X ⊕ X_{d}^{−1} ⊕ {e})^{2} → AP_{2}(X) is a closed mapping.

Proof. Obviously, we have (3) ⇒ (1) and (4) ⇒ (2). Moreover, it follows from Theorems 3.6 and 3.7 that we have (2) ⇒ (1). It suffices to show that (1) ⇒ (3) and (2) ⇒ (4).

(1) ⇒ (3). Clearly, both F P_{2}(X) ∖ F P_{1}(X) and F P_{1}(X) ∖ {e} are open in F P_{2}(X). Let E be a closed subset in (X ⊕ X_{d}^{−1} ⊕ {e})^{2}. To show that i_{2}(E) is closed in F P_{2}(X) take w ∈ \overline{i_{2}(E)}.

Case a1: w ∈ F P_{1}(X)∖{e}. Suppose w ∉ i_{2}(E), then (w, e) ∉ E and (e, w) ∉ E. Since E is closed, there is an open neighborhood U (open in X ∪ X_{d}^{−1} ) of w such that (U × {e}) ∩ E = ∅ and ({e} × U) ∩ E = ∅. Obviously, we have (U ×{e})∪({e}×U) = i_{2}^{−1} (U). Then U is open in F P_{2}(X) since (U ×{e})∪({e}×U) is open in (X ⊕ X_{d}^{−1} ⊕ {e})^{2} and i_{2} is a quotient map. Hence U ∩ i_{2}(E) = ∅, which contradicts w ∈ \overline{i_{2}(E)}.

Case a2: w ∈ F P_{2}(X) ∖ F P_{1}(X). Let w = w_{1}^{ϵ_{1}}w_{2}^{ϵ_{2}}, where w_{i} ∈ X and ϵ_{i} = 1 or -1 (i = 1, 2). Suppose that w ∉ i_{2}(E). Then (w_{1}^{ϵ_{1}}, w_{2}^{ϵ_{2}}) ∉ E.

Subcase a21: ϵ_{1} = ϵ_{2} = 1. Since (w_{1}, w_{2}) ∉ E and E is closed in (X ⊕ X_{d}^{−1} ⊕ {e})^{2}, there exist neighborhoods U and V of w_{1} and w_{2} in X, respectively, such that (U × V) ∩ E = ∅. Therefore, it is easy to see that UV ∩ i_{2}(E) = ∅. From Theorem 3.2 it follows that UV is a neighborhood of w, hence w ∉ \overline{i_{2}(E)}, which is a contradiction.

Subcase a22: ϵ_{1} = ϵ_{2} = −1. From Theorem 3.2 it follows that {w_{1}^{−1} w_{2}^{−1} } is a neighborhood of w, then w ∉ \overline{i_{2}(E)}, which is a contradiction.

Subcase a23: ϵ_{1} ≠ ϵ_{2}. Without loss of generality, we may assume that ϵ_{1} = 1 and ϵ_{2} = −1. Then since (w_{1}, w_{2}^{−1} ) ∉ E and E is closed in (X ⊕X_{d}^{−1} ⊕{e})^{2}, there exists a neighborhood U of w_{1} in X such that (U × {w_{2}^{−1} }) ∩ E = ∅ and w_{2} ∉ U. (This is possible since X is T_{1}.) Obviously, Uw_{2}^{−1} ⊂ F P_{2}(X)∖F P_{1}(X). Therefore, it is easy to see that Uw_{2}^{−1} ∩ i_{2}(E) = ∅. From Theorem 3.2 it follows that Uw_{2}^{−1} is a neighborhood of w, hence w ∉ \overline{i_{2}(E)}, which is a contradiction.

Therefore, we have w ∈ i_{2}(E).

Case a3: w = e. Suppose that e ∉ i_{2}(E). Then E ∩ (Δ_{1} ∪ Δ_{2} ∪ {(e, e)}) = ∅.

For any x ∈ X, since E does not contain points (x^{−1} , x) and (x, x^{−1} ), there exists an open neighborhood U(x) of x in X such that ({x^{−1} } × U(x)) ∩ E = ∅ and (U(x) × {x^{−1} }) ∩ E = ∅. Let U = ⋃ _{x∈X} ({x^{−1} } × U(x)) and V = ⋃ _{x∈X} (U(x) × {x^{−1} }). Then U ∩ E = ∅ and V ∩ E = ∅. Let W = U ∪ V ∪ {e} × {e}. Then W is open in (X ⊕ X_{d}^{−1} ⊕ {e})^{2} by (2) of Theorem 3.7. Obviously, we have W ∩ E = ∅. It is easy to see that i_{2}^{−1} (i_{2}(W)) = W, then i_{2}(W) is open since i_{2} is a quotient map. Hence i_{2}(W) ∩ i_{2}(E) = ∅, this is a contradiction.

(2) ⇒ (4). (Note: The proof is almost similar to (1) ⇒ (3). However, we give out the proof for the convenience of readers.) Clearly, both AP_{2}(X) ∖ AP_{1}(X) and AP_{1}(X) ∖ {e} are open in AP_{2}(X). Let E be a closed subset in (X ⊕ −X_{d} ⊕ {e})^{2}. To show that i_{2}(E) is closed in AP_{2}(X) take w ∈ \overline{i_{2}(E)}.

Case b1: w ∈ AP_{1}(X)∖{e}. Suppose w ∉ i_{2}(E), then (w, e) ∉ E and (e, w) ∉ E. Since E is closed, there is an open neighborhood U (open in X ∪ −X_{d}) of w such that (U × {e}) ∩ E = ∅ and ({e} × U) ∩ E = ∅. Obviously, we have (U ×{e})∪({e}×U) = i_{2}^{−1} (U). Then U is open in AP_{2}(X) since (U ×{e})∪({e}×U) is open in (X ⊕ −X_{d} ⊕ {e})^{2} and i_{2} is a quotient map by Theorems 3.6 and 3.7. Then U ∩ i_{2}(E) = ∅, that contradicts w ∈ \overline{i_{2}(E)}.

Case b2: w ∈ AP_{2}(X) ∖ AP_{1}(X). Let w = ϵ_{1}w_{1} + ϵ_{2}w_{2}, where w_{i} ∈ X and ϵ_{i} = 1 or -1 (i = 1, 2). Suppose that w ∉ i_{2}(E). Then (ϵ_{1}w_{1}, ϵ_{2}w_{2}) ∉ E and (ϵ_{2}w_{2}, ϵ_{1}w_{1}) ∉ E.

Subcase b21: ϵ_{1} = ϵ_{2} = 1. Since {(w_{1}, w_{2}), (w_{2}, w_{1})} ∉ E and E is closed in (X ⊕ −X_{d} ⊕ {e})^{2}, there exist neighborhoods U and V of w_{1} and w_{2} in X, respectively, such that (U × V ∪ V × U) ∩ E = ∅. Therefore, it is easy to see that (U + V) ∩ i_{2}(E) = ∅. From Theorem 3.3 it follows that U + V is a neighborhood of w, hence w ∉ \overline{i_{2}(E)}, which is a contradiction.

Subcase b22: ϵ_{1} = ϵ_{2} = −1. From Theorem 3.2 it follows that {−w_{1} − w_{2}} is a neighborhood of w, then w ∉ \overline{i_{2}(E)}, which is a contradiction.

Subcase b23: ϵ_{1} ≠ ϵ_{2}. Without loss of generality, we may assume that ϵ_{1} = 1 and ϵ_{2} = −1. Then since {(w_{1}, −w_{2}), (−w_{2}, w_{1})} ∉ E and E is closed in (X ⊕ −X_{d} ⊕ {e})^{2}, there exists a neighborhood U of w_{1} in X such that (U × {w_{2}^{−1} } ∪ {w_{2}^{−1} } × U) ∩ E = ∅ and w_{2} ∉ U. (This is possible since X is T_{1}.) Obviously, U −w_{2} ⊂ AP_{2}(X)∖AP_{1}(X). Therefore, it is easy to see that (U −w_{2})∩i_{2}(E) = ∅. From Theorem 3.3 it follows that U − w_{2} is a neighborhood of w, hence w ∉ \overline{i_{2}(E)}, which is a contradiction.

Therefore, we have w ∈ i_{2}(E).

Case b3: w = e. Suppose that e ∉ i_{2}(E). Then E ∩ (Δ_{1} ∪ Δ_{2} ∪ {(e, e)}) = ∅.

For any x ∈ X, since E does not contain points (−x, x) and (x, −x), there exists an open neighborhood U(x) of x in X such that ({−x} × U(x)) ∩ E = ∅ and (U(x)×{−x})∩E = ∅. Let U = ⋃ _{x∈X} ({−x}×U(x)) and V = ⋃ _{x∈X} (U(x)×{−x}). Then U ∩ E = ∅ and V ∩ E = ∅. Let W = U ∪ V ∪ {e} × {e}. Then W is open in (X ⊕ −X_{d} ⊕ {e})^{2} by Theorem 3.7. Obviously, we have W ∩ E = ∅. It is easy to see that i_{2}^{−1} (i_{2}(W)) = W, then i_{2}(W) is open in AP_{2}(X) since i_{2} is a quotient map by Theorems 3.6 and 3.7. Hence i_{2}(W) ∩ i_{2}(E) = ∅, which is a contradiction. □

Proposition 3.1. Let X be a T_{1}-space. Then, for some n ⩾ 3,

i_{n} : (X ⊕ X_{d}^{−1} ⊕ {e})^{n} → F P_{n}(X)

is a closed map if and only if X is discrete.

Proof. If X is discrete, then F P (X) is discrete, so it is easy to see that each i_{n} is a closed map.

Let i_{n} be a closed map for some n ⩾ 3. Since X is T_{1}, then X^{−1} is discrete. Suppose that X is not discrete, then there exists x ∈ X such that x ∈ \overline{X ∖ \{x\}}. Let

A = {(x_{α} , x_{α} , x_{α}^{−1} , e, …, e) ∈ (X ⊕ X_{d}^{−1} ⊕ {e})^{n} : x_{α} ∈ X ∖ {x}}.

Then A is a closed discrete subset of (X ⊕ X_{d}^{−1} ⊕ {e})^{n}, and therefore, i_{n}(A) = X∖{x} is a closed discrete subset, which is a contradiction. Hence X is discrete. □

Note. Therefore, we can improve all results in [3, Sections 4 and 5] from quotient mappings to closed mappings. For example, we have the following proposition.

Proposition 3.2. The mapping i_{2} is a closed mapping for any countable T_{1}- space. In particular, the mapping i_{2} is a closed mapping for any countable subspace of the real line ℝ.

Corollary 3.1. F P_{2}(ℚ) and AP_{2}(ℚ) are Fréchet, where ℚ is the rational number of real line ℝ.

Proof. By Proposition 3.2, i_{2} is a closed mapping. Then F P_{2}(ℚ) and AP_{2}(ℚ) are Fréchet since (X ⊕ X_{d}^{−1} ⊕ {e})^{2} is Fréchet and closed mappings preserve the property of Fréchet. □

By [5, Proposition 6.26], we also have the following proposition.

Proposition 3.3. For an arbitrary compact first-countable Hausdorff space X, the mapping i_{2} is closed if and only if X is countable.

Acknowledgements. We wish to thank the reviewers for the detailed list of corrections, suggestions to the paper, and all her/his efforts in order to improve the paper. The first author is supported by the NSFC (Nos. 11571158, 11201414, 11471153) the Natural Science Foundation of Fujian Province (No. 2017J01405) of China, the Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA13190) and the foundation of The Education Department of Fujian Province(No. JA14200).

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School of Mathematics and Statistics (Received 24 04 2014) Minnan Normal University (Revised 14 10 2014) Zhangzhou P. R. China

linfucai2008@aliyun.com

Department of Mathematics Ohio University Zanesville Campus Zanesville USA

liuc1@ohio.edu