@ Appl. Gen. Topol. 16, no. 2(2015), 89-98doi:10.4995/agt.2015.1874
c© AGT, UPV, 2015

Free paratopological groups

Ali Sayed R. Elfard

School of Mathematics and Applied Statistics, University of Wollongong, Australia (a.elfard@yahoo.com)

Abstract

Let FP(X) be the free paratopological group on a topological space
X in the sense of Markov. In this paper, we study the group FP(X)
on a Pα-space X where α is an infinite cardinal and then we prove

that the group FP(X) is an Alexandroff space if X is an Alexandroff
space. Moreover, we introduce a neighborhood base at the identity of

the group FP(X) when the space X is Alexandroff and then we give
some properties of this neighborhood base. As applications of these, we

prove that the group FP(X) is T0 if X is T0, we characterize the spaces
X for which the group FP(X) is a topological group and then we give
a class of spaces X for which the group FP(X) has the inductive limit
property.

2010 MSC: Primary 22A30; secondary 54D10; 54E99; 54H99.

Keywords: Topological group; paratopological group; free paratopological

group; Alexandroff space; partition space, neighborhood base at

the identity.

1. Introduction

Let FP(X) and AP(X) be the free paratopological group and the free
abelian paratopological group, respectively, on a topological space X in the
sense of Markov. The group FP(X) is the abstract free group Fa(X) on X
with the strongest paratopological group topology on Fa(X) that induces the
original topology on X and the abelian group AP(X) is the abstract free
abelian group Aa(X) on X with the strongest paratopological group topol-
ogy on Aa(X) that induces the original topology on X. For more information
about free paratopological groups, see ([11], [7], [3], [4], [5]).

Received 13 November 2013 – Accepted 8 March 2015

http://dx.doi.org/10.4995/agt.2015.1874


A. Elfard

In 1937, P. Alexandroff [10] introduced a class of topological spaces under
the name of Diskrete Räume (Discrete space) which is a space in which an
arbitrary intersections of open sets is open. Now the name has been changed
to Alexandroff space since the discrete space is a space in which every sin-
gleton set is open. Recently, researchers have shown an increased interest in
studying Alexandroff spaces. This may be due to the important applications
of Alexandroff spaces in some areas of mathematical sciences such as the field
of computer science.

In this paper, we study the groups FP(X) and AP(X) on a Pα-space X,
where α is an infinite cardinal (a topological space X is a Pα-space, where α
is an infinite cardinal if the set

⋂

C is open in X for each family C of open
subsets of X with |C | < α). Then in Theorem 4.1, we prove that the groups
FP(X) and AP(X) are Alexandroff spaces if the space X is Alexandroff and in
Theorem 4.4, we introduce simple neighborhood bases at the identities of the
groups FP(X) and AP(X) for their topologies. Moreover, we study the groups
FP(X) and AP(X) in the case where X is a partition space and in another
case where X is a T0 Alexandroff space.

As applications of these results, in Theorem 5.1, we characterize the spaces
X for which the paratopological groups FP(X) and AP(X) are topological
groups and in Theorem 5.6, we prove that the group FP(X) is T0 if the space
X is T0. Finally, in Theorem 5.7, we give a class of spaces X for which the
groups FP(X) and AP(X) have the inductive limit property.

The content of this paper is adapted from the author’s thesis [5], chapter
3. We remark that the results in Theorem 5.1 and Theorem 5.6 were found
independently by the author in his thesis [5]. However, similar to these results
were found by Pyrch ([8], [9]).

2. Definitions and Preliminaries

A paratopological group is a pair (G, T ), where G is a group and T is a
topology on G such that the mapping (x, y) 7→ xy of G×G into G is continuous.
If in addition, the mapping x 7→ x−1 of G into G is continuous, then (G, T ) is
a topological group.

If (G, T ) is a paratopological group, then simply we denote it by G.

Marin and Romaguera [6] described a complete neighborhood base at the
identity of any paratopological group as follows:

Proposition 2.1. Let G be a group and let N be a collection of subsets of G,
where each member of N contains the identity element e of G. Then the
collection N is a base at e for a paratopological group topology on G if and
only if the following conditions are satisfied:

(1) for all U, V ∈ N , there exists W ∈ N such that W ⊆ U ∩ V ;
(2) for each U ∈ N , there exists V ∈ N such that V 2 ⊆ U;
(3) for each U ∈ N and for each x ∈ U, there exists V ∈ N such that

xV ⊆ U and V x ⊆ U; and

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Free paratopological groups

(4) for each U ∈ N and each x ∈ G, there exists V ∈ N such that
xV x−1 ⊆ U.

Definition 2.2 ([3]). Let X be a subspace of a paratopological group G.
Suppose that

(1) the set X generates G algebraically, that is, 〈X〉 = G and
(2) every continuous mapping f : X → H of X to an arbitrary paratopo-

logical group H extends to a continuous homomorphism f̂ : G → H.

Then G is called the Markov free paratopological group on X, and is denoted
by FP(X).
By substituting “abelian paratopological group” for each occurrence of “paratopo-
logical group” above we obtain the definition of the Markov free abelian paratopo-
logical group on X and we denote it by AP(X).

Remark 2.3. We denote the free topology of FP(X) by TF P and the free topol-
ogy of AP(X) by TAP and we note that the topologies TF P and TAP are the
strongest paratopological group topologies on the underlying sets of FP(X)
and AP(X), respectively, that induce the original topology on X.

3. Pα-spaces

Let X be a topological space and α be an infinite cardinal. We say that X
is a Pα-space if the set

⋂

C is open in X for each family C of open subsets of
X with |C | < α. Let τ be the topology of X. Then we define the topology τα
to be the intersection of all topologies O on X where τ ⊆ O and (X, O) is
a Pα-space. Since the discrete topology on X contains τ and is a Pα-space, τα
exists and (X, τα) is a Pα-space. We call the topology τα the Pα-modification
of τ.

We note that if X is a Pα-space, then X is a Pα+-space, where α
+ is the

successor cardinal of α.

For the remain of this section we assume that α is a fixed infinite cardinal
unless we say otherwise.

Theorem 3.1. Let (X, τ) be a topological space and let α+ be the infinite
successor cardinal of α. Then the collection of all sets which are the intersection
of fewer than β open subsets of X is a base for the topology τα on X, where
β = α if α is regular and β = α+ if α is singular.

Proof. Let τ = {Ui}i∈I. We show that the collection B = {
⋂

d∈D Ud : D ⊆ I
and |D| < β} of subsets of X is a base for the topology τα on X, where β as
defined in the statement of the theorem. It is well known that every infinite
successor cardinal is regular, so in both cases, β is regular.

We show first that B is a base for some topology τ∗ on X. If x ∈ X, then
there exists i0 ∈ I where x ∈ Ui0 and such that Ui0 ∈ B. Let B1, B2 ∈ B and
let x ∈ B1 ∩B2. Assume that B1 =

⋂

d∈D Ud and B2 =
⋂

t∈T Ut, where D, T ⊆

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A. Elfard

I and |D|, |T | < β. Let R = D ∪ T . So |R| < β. Hence B3 =
⋂

r∈R Ur ∈ B
and x ∈ B3 ⊆ B1 ∩ B2. Therefore, B is a base for some topology τ

∗ on X.
We show second that (X, τ∗) is a Pα-space. Let τ

∗ = {Vj}j∈J and let M ⊆ J
where |M| < β. Then we have

⋂

m∈M

Vm =
⋂

m∈M

⋃

i∈Im

Bm,i

=
⋃

f : M→
⋃

m∈M
Im,f(m)∈Im∀m∈M

(
⋂

m∈M

Bm,f(m)) ∈ τ
∗,

where Im is an index set and Bm,i ∈ B for all m ∈ M and i ∈ Im. Thus τ
∗

contains τ and (X, τ∗) is a Pβ-space, which implies that in both cases of β,
(X, τ∗) is a Pα-space.

Now let τ̂ be a topology on X containing τ such that (X, τ̂) is a Pα-space.
Then in the case where α is regular, we have B ⊆ τ̂ and in the case where
α is singular, by the argument above, we have (X, τ̂) is a Pα+-space, which
implies that B ⊆ τ̂. Thus τ∗ ⊆ τ̂ and hence τ∗ is the smallest topology on X
containing τ such that (X, τ∗) is a Pα-space. Therefore, τ

∗ = τα. �

Proposition 3.2. Let (G, τ) be a paratopological group. Then (G, τα) is a
paratopological group.

Proof. Let g1, g2 ∈ G and let U ∈ τα contains g1g2. By Theorem 3.1, there is a
set Λ, where |Λ| < β and β is as in the theorem such that g1g2 ∈

⋂

λ∈Λ Uλ ⊆ U
where Uλ ∈ τ for all λ ∈ Λ. Thus g1g2 ∈ Uλ for all λ ∈ Λ. Since τ is a
paratopological group topology on G, for each λ ∈ Λ, there are V (λ), W(λ) ∈ τ
containing g1, g2, respectively, such that V (λ)W(λ) ⊆ Uλ. Let U1 =

⋂

λ∈Λ V (λ)
and U2 =

⋂

λ∈Λ W(λ). Then U1U2 ⊆ Uλ for all λ ∈ Λ. Hence, U1, U2 ∈ τα and
U1U2 ⊆

⋂

λ∈Λ Uλ ⊆ U. Therefore, τα is a paratopological group topology on
G. �

Proposition 3.3. Let X be a topological space. Then the group FP(X) is a
Pα-space if and only if the space X is a Pα-space.

Proof. =⇒: It is easy to prove that X is a Pα-space.
⇐=: Let τ be the topology of X and let TF P be the free topology of FP(X).

We show that (TF P )α = TF P . By Proposition 3.2, (TF P )α is a paratopological
group topology on Fa(X) and it is stronger than TF P . However, TF P is the free
paratopological group topology on Fa(X), which is the strongest paratopolog-
ical group topology on Fa(X) inducing the original topology τ on X. Since
(TF P )α|X = (TF P |X)α and (TF P |X)α = (τ)α = τ, (TF P )α induces the topology
τ of X. Thus we have (TF P )α = TF P and therefore, FP(X) is a Pα-space. �

The same result of Proposition 3.3 is true for AP(X).

4. Free paratopological groups on Alexandroff spaces

A topological space X is said to be Alexandroff [1] if the intersection of
every family of open subsets of X is open in X.

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Free paratopological groups

We note that a topological space X is Alexandroff if and only if X is a Pα-
space for every infinite cardinal α. Thus by using Proposition 3.3, we get the
next result.

Theorem 4.1. The group FP(X) (AP(X)) on a space X is an Alexandroff
space if and only if X is an Alexandroff space.

Let G be a group and let H be a submonoid of G. Then we say that H is a
normal submonoid of G if ghg−1 ∈ H for all h ∈ H and g ∈ G.

Proposition 4.2. If H is a normal submonoid of a group G, then {H} is a
neighborhood base at the identity of G for a paratopological group topology on
G.

Proof. Let H be a normal submonoid of G. Then {H} satisfies the conditions
of Proposition 2.1, and therefore, {H} is a neighborhood base at the identity
of G for a paratopological group topology on G. �

Let X be a topological space. For each x ∈ X, let U(x) =
⋂

{U : x ∈ U and
U is open}. Then it is easy to see that the space X is Alexandroff if and only
if the set U(x) is open in X for each x ∈ X.

Let X be an Alexandroff space and let FP(X) and AP(X) be the free
paratopological group and the free abelian paratopological group, respectively,
on X. Let UA =

⋃

x∈X(U(x) − x) ⊆ AP(X). Then we define NA to be the
smallest submonoid of AP(X) containing the set UA. So NA is of the form

NA = {y1 − x1 + y2 − x2 + · · · + yn − xn : xi ∈ X, yi ∈ U(xi) for all
i = 1, 2, . . . , n, n ∈ N}.

Or simply, we write NA = 〈UA〉. Since every submonoid of an abelian group
is normal, NA is normal. However, in this case, we will omit the word normal
and say submonoid.

Let NA = {NA}. Since NA is a submonoid of AP(X), by Proposition 4.2,
NA is a neighborhood base at the identity 0A of AP(X) for a paratopological
group topology OA on Aa(X).

Now for the group FP(X), let UF =
⋃

x∈X x
−1U(x) ⊆ FP(X) and then we

define NF to be the smallest normal submonoid of FP(X) containing the set
UF . The normal submonoid NF consists exactly of the set of all elements of
the form,

w = g1x
−1
1 y1g

−1
1 · g2x

−1
2 y2g

−1
2 · · · gnx

−1
n yng

−1
n

where n ∈ N, g1, g2, . . . , gn is an arbitrary finite system of elements of Fa(X)
and x−11 y1, x

−1
2 y2, . . . , x

−1
n yn is an arbitrary finite system of elements of UF .

Define NF = {NF }. By Proposition 4.2, NF is a neighborhood base at the
identity e of FP(X) for a paratopological group topology OF on the free group
Fa(X).

Proposition 4.3. The topologies OF and OA induce topologies coarser than
the original topology on X.

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A. Elfard

Theorem 4.4. The collection NF (NA) is a neighborhood base at e (0A) for
the free topology of FP(X) (AP(X)).

Proof. We prove the theorem for NF , since the proof for NA is the same. We
show first that the topology OF is finer than the free topology TF P of FP(X).
Let ξ : X → G be a continuous mapping of the space X into an arbitrary
paratopological group G. Then ξ extends to a homomorphism ξ̂ : Fa(X) → G.

We show that ξ̂ is continuous with respect to the topology OF . Let V be a
neighborhood of ξ̂(e) = eG in G. Fix x ∈ X. Then ξ(x)V is a neighborhood

of ξ(x) in G. Since ξ is continuous at x, ξ(U(x)) ⊆ ξ(x)V and Since ξ̂|X = ξ,

ξ̂(U(x)) ⊆ ξ̂(x)V . Because ξ̂ is a homomorphism, ξ̂
(

x−1U(x)
)

⊆ V . Since x is
any point in X, we have

(4.1) ξ̂
(

⋃

x∈X

x−1U(x)
)

⊆ V.

Fix n ∈ N. Then there exists a neighborhood U of eG in G such that U
n ⊆ V

and also, for all g ∈ Fa(X), there exists a neighborhood W of eG in G such

that ξ̂(g)W
(

ξ̂(g)
)−1

⊆ U. Since V is any neighborhood of eG in G, from (4.1),

we have ξ̂
(
⋃

x∈X x
−1U(x)

)

⊆ W . Fix g ∈ Fa(X). So we have

ξ̂(g)ξ̂
(

⋃

x∈X

x−1U(x)
)(

ξ̂(g)
)−1

⊆ ξ̂(g)Wξ̂(g)−1.

Since ξ̂ is a homomorphism,

(4.2) ξ̂
(

⋃

x∈X

gx−1U(x)g−1
)

⊆ ξ̂(g)Wξ̂(g)−1 ⊆ U.

Since (4.2) holds for every g ∈ Fa(X), we have

ξ̂
(

⋃

g∈Fa(X)

⋃

x∈X

gx−1U(x)g−1
)

⊆ U.

Thus we have

ξ̂
(

(

⋃

g∈Fa(X)

⋃

x∈X

gx−1U(x)g−1
)n
)

⊆ Un ⊆ V.

Since n is any element of N,

ξ̂
(

⋃

n∈N

(

⋃

g∈Fa(X)

⋃

x∈X

gx−1U(x)g−1
)n
)

⊆ V.

Since NF =
⋃

n∈N

(
⋃

g∈Fa(X)

⋃

x∈X gx
−1U(x)g−1

)n
, we have ξ̂(NF ) ⊆ V .

Thus ξ̂ is continuous with respect to the topology OF and therefore, OF is
finer than TF P . By Proposition 4.3, OF |X is coarser than the original topol-
ogy on X. Since OF is finer that TF P , OF |X induces the original topology
on X. Thus we satisfied the conditions of Definition 2.2, which implies that
OF = TF P . Therefore, NF is a neighborhood base at e of the group FP(X). �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 94



Free paratopological groups

Now let HF = {gNF : g ∈ Fa(X)} and let HA = {g + NA : g ∈ Aa(X)}. If
g1, g2 ∈ FP(X) such that g1 ∈ g2NF , then we have g1NF ⊆ g2NF NF = g2NF .
A similar result is true for HA.

Let X be a set. Then for all k ∈ Z, we define Zk(X) = {x
ǫ1
1 x

ǫ2
2 . . . x

ǫn
n ∈

Fa(X) :
∑n

i=1 ǫi = k} and Z
A
k (X) = {ǫ1x1 + ǫ2x2 + · · · + ǫnxn ∈ Aa(X) :

∑n

i=1 ǫi = k}. For every k1, k2 ∈ Z and k1 6= k2, the sets Zk1(X) and Zk2(X)
are disjoint and the sets ZAk1(X) and Z

A
k2
(X) are disjoint. The set Z0(X) is

the smallest normal subgroup of Fa(X) containing the set ZF =
⋃

x∈X x
−1X

and the set ZA0 (X) is the smallest subgroup of Aa(X) containing the set
ZA =

⋃

x∈X X − x.

Let X be a topological space and let I : X → AP(X) be the identity map-
ping of the space X to the abelian group AP(X). Then we extend I to the

continuous homomorphism mapping Î : FP(X) → AP(X). We call the map-

ping Î the canonical mapping.

Theorem 4.5. Let X be an Alexandroff space. Then the following are equiv-
alent.

(1) The space X is indiscrete.
(2) NF = Z0(X) in FP(X).
(3) NA = Z

A
0 (X) in AP(X).

Proof. (1)⇒(2): Assume that X is indiscrete. Then U(x) = X for all x ∈ X
and so UF = ZF , where ZF is the generating set for Z0(X). Therefore, NF =
Z0(X).

(2)⇒(3): Assume that NF = Z0(X). Let Î : FP(X) → AP(X) be the

canonical mapping. Thus Î(NF ) = Î(Z0(X)). Since Î(NF ) = NA and Î(Z0(X)) =
ZA0 (X), so NA = Z

A
0 (X).

(3)⇒(1): Assume that NA = Z
A
0 (X). Thus Z

A
k
(X) is open in AP(X) for

each k ∈ Z. Since ZA1 (X) ∩ X = X and Z
A
k (X) ∩ X = ∅ for all k ∈ Z \ {1},

we have X is indiscrete. �

We call a space X a partition space if X has a base which is a partition of
X. Clearly, every partition space is an Alexandroff space.

It is easy to see that if X is a partition space, then the collection {U(x)}x∈X
is a partition on X.

Theorem 4.6. If X is a partition space, then the free paratopological groups
FP(X) and AP(X) are partition spaces.

Proof. Let X be a partition space. Then NF and NA are normal subgroups of
FP(X) and AP(X), respectively. Thus the collections HF and HA as defined
above are partitions of FP(X) and AP(X), respectively. Therefore the result
follows. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 95



A. Elfard

5. Applications

Let TA be the topology of the subspace X
−1 of FP(X), where X be any

topological space. Then by Theorem 4.2 of [3], the topology TA has as an open
base the collection {C−1 : C closed in X}. In this topology, the intersection
of every collection of open subsets is open, and the space X−1

A
= (X−1, TA) is

therefore an Alexandroff space.

Theorem 5.1. Let X be a topological space. Then the group FP(X) on X is
a topological group if and only if X is a partition space.

Proof. =⇒: Assume that FP(X) is a topological group. Let U be an open set
in X. By the argument above, the topology on the subspace X−1

A
of FP(X)

has the collection {C−1 : C is closed in X} as a base. Thus (Uc)−1 is open in
X−1

A
. Since the inversion mapping of X−1 to X is a homeomorphism, Uc is

open in X. So U is closed in X and therefore, X is a partition space.
⇐=: If X is a partition space, then NF is a subgroup of FP(X). Therefore,

FP(X) is a topological group.
The same proof works for AP(X). �

Proposition 5.2. Let X be an Alexandroff space and let FP(X) be the free
paratopological group on X. Then the space X is T0 if and only if for each
w ∈ NF and w 6= e we have Î(w) 6= 0A.

Proof. =⇒: Suppose that there exists w ∈ NF and w 6= e such that Î(w) = 0A,
where

w = g1x
−1
1 y1g

−1
1 g2x

−1
2 y2g

−1
2 · · · gnx

−1
n yng

−1
n for some n ∈ N, yi 6= xi and

yi ∈ U(xi) for all i = 1, 2, . . . , n.

Then we have Î(w) = y1 − x1 + y2 − x2 + · · · + yn − xn = 0A. If n = 1, then

x1 = y1, which gives a contradiction. Assume that n > 1. Since Î(w) = 0A,
for each i ∈ A = {1, 2, . . . , n}, there exists ji ∈ A, where ji 6= i such that
xi = yji. Define σ : A → A by setting σ(i) = ji for all i ∈ A. Clearly σ is a
permutation on A. Since any permutation can be written as product of cycles,
there are m ∈ N, where 2 ≤ m ≤ n and distinct i1, i2, . . . , im ∈ A such that
σ(i1) = i2, σ(i2) = i3, . . . , σ(im−1) = im, σ(im) = i1 and such that xik = yσ(ik)
for k = 1, 2, . . . , m. Thus xi1 = yi2, xi2 = yi3, . . . , xim−1 = yim, xim = yi1 and
hence

U(yi1) ⊆ U(xi1) = U(yi2) ⊆ U(xi2 ) = U(xi3) ⊆ · · ·

⊆ U(xim−1) = U(yim) ⊆ U(xim) = U(yi1),

which implies that

U(yi1) = U(xi1 ) = U(yi2) = U(xi2 ) = · · · = U(xim−1 ) = U(yim).

Thus we can not separate the points yi1, xi1, yi2, xi2, . . . , xim−1, yim. Therefore,
X is not a T0 space.

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Free paratopological groups

⇐=: Assume that X is not T0. Then there are x, y ∈ X such that x 6= y
and U(x) = U(y), which implies that x ∈ U(y) and y ∈ U(x). Hence

w = x−1yxy−1 = (x−1y)(x(y−1x)x−1) ∈ NF

and w 6= e, but Î(w) = 0A. Therefore, the space X is T0. �

We note that a corollary of this result is that if X is an Alexandroff T0 space,
then Î has the property that ker Î ∩ NF = {e}.

The following result is easy to prove.

Proposition 5.3. Let G be a paratopological group. Then G is a T0 space if
and only if for all a ∈ G such that a 6= e, there exists a neighborhood U of e
such that either a /∈U or a−1 /∈ U.

Proposition 5.4. Let X be an Alexandroff space. Then FP(X) is a T0 space
if and only if X is a T0 space.

Proof. =⇒: Since X is a subspace of FP(X), the result follows.
⇐=: Assume that X is T0. We claim that FP(X) is T0. In fact, if FP(X)

is not T0, then by Proposition 5.3, there exists w ∈ FP(X), w 6= e such that

w, w−1 ∈ NF . Hence by Proposition 5.2, Î(w) 6= 0A and it is easy to see that

Î(w), −Î(w) ∈ NA, which implies that Î(w), −Î(w) are in every neighborhood
of 0A. Once again by Proposition 5.3, AP(X) is not T0 and so by Proposition
3.4 of [7] (which says that if a space X is T0, then AP(X) is T0), X is not a
T0 space, which gives a contradiction. Therefore, FP(X) is T0. �

Fix n ∈ N and let Rn = {1, 2, . . . , n} ⊆ N. For i = 0, 1, . . . , n, define
Rn,i = {1, 2, . . . , i} and τn = {Rn,i : i = 0, . . . , n}. Then it is easy to see that
τn is a topology on Rn. Let m, k ∈ Rn, where m 6= k and assume that m < k.
Then m ∈ Rn,m and k /∈ Rn,m. Therefore, (Rn, τn) is a T0 space.

Proposition 5.5. Let X be a T0 space and let x1, x2, . . . , xn be distinct ele-
ments of X. Then there exists a continuous mapping µ: X → Rn such that
µ|{x1,x2,...,xn} is one-to-one.

Theorem 5.6. Let X be a topological space. Then the free paratopological
group FP(X) on X is T0 if and only if the space X is T0.

Proof. =⇒: It is clear.
⇐=: Let w = xǫ11 x

ǫ2
2 · · · x

ǫm
m ∈ FP(X) for some m ∈ N such that w 6= e.

Choose indices i1, i2, . . . , in for some n ≤ m such that xi1 , xi2, . . . , xin are
the distinct letters among x1, x2, . . . , xm. Then by Proposition 5.5, there ex-
ists a continuous mapping µ : X → Rn such that µ|{xi1 ,xi2 ,...,xin } is one-to-
one, where Rn is the space defined above. Then we extend µ to a continu-
ous homomorphism µ̂ : FP(X) → FP(Rn). Since µ|{xi1 ,...,xin } is one-to-one,
µ̂(w) = [µ̂(x1)]

ǫ1[µ̂(x2)]
ǫ2 · · · [µ̂(xn)]

ǫn 6= e∗, where e∗ is the identity of FP(Rn).
By Proposition 5.4, we have FP(Rn) is a T0 space. So there is an open set U
in FP(Rn), which contains one of e

∗ or µ̂(w) and does not contain the other.
Say e∗ ∈ U and µ̂(w) /∈ U. Since µ̂ is continuous, µ̂−1(U) is an open set in

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 97



A. Elfard

FP(X) such that e ∈ µ̂−1(U) and w /∈ µ̂−1(U). Similarly for the other case.
Therefore, the free paratopological group FP(X) is T0. �

A topological space X is said to be the inductive limit of a cover C of X if
a subset V of X is open whenever V ∩ U is open in U for each U ∈ C .

A parallel result of the next theorem was proved in Proposition 7.4.8 of [2]
in the case of free topological groups.

Theorem 5.7. Let X be a T1 P-space. Then the free paratopological group
FP(X) (AP(X)) is the inductive limit of the collection {FPn(X): n ∈ N}
({APn(X): n ∈ N}).

Proof. We prove the statement for FP(X), since the proof for AP(X) is similar.
Let C be a subset of FP(X) such that C ∩ FPn(X) is closed in FPn(X) for all
n ∈ N. By Theorem 4.1.3 of [3], the sets FPn(X) are closed in FP(X) for all
n ∈ N. Thus the sets C ∩ FPn(X) are closed in FP(X) for all n ∈ N, which
implies that C is a countable union of closed sets in FP(X). Since the group
FP(X) is a P-space, C is closed in FP(X) and then FP(X) is the inductive
limit of the collection {FPn(X): n ∈ N}. �

Acknowledgements. The author wishes to thank associate professor Peter

Nickolas for helpful comments.

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