@ Appl. Gen. Topol. 15, no. 1 (2014), 25-32doi:10.4995/agt.2014.2049
c© AGT, UPV, 2014

Near metrizability via a new approach

D. Mandal
∗,a

and M. N. Mukherjee
a

a
Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road,

Kolkata–700019, India (dmandal.cu@gmail.com, mukherjeemn@yahoo.co.in)

Abstract

The present article deals with near metrizability, initiated in an ear-

lier paper [7], with a new orientation and approach. The notions of
nearly regular and uniform pseudo-bases are introduced and analogues

of some results concerning metrizability and paracompactness are ob-

tained for near metrizability and near paracompactness respectively via

the proposed approach, suitably formulated.

2010 MSC: 54D20; 54E99.

Keywords: nearly paracompact space; regular open set; nearly regular and

uniform pseudo-bases; nearly metrizable.

1. Introduction

The idea of near paracompactness, a well known weaker form of paracom-
pactness, was initiated by Singal and Arya [9], followed by an extensive study of
the concept by many topologists from different perspectives and with different
applications (for instance see [3], [4], [5], [6], [8]). Now, in [7] we introduced a
neighbouring form of metrizability, termed near metrizability, which plays the
same role with regard to near paracompactness as is done by metrizability vis-
a-vis paracompactness. It was shown in [7] that there exist nearly metrizable,
non-metrizable spaces that are not paracompact, moreover some other facts
were established in [7].

∗The author is thankful to the University Grants Commission, New Delhi- 110002, In-
dia for sponsoring this work under Minor Research Project vide letter no. F. No. 41-
1388/2012(SR).

Received November 2012 – Accepted July 2013

http://dx.doi.org/10.4995/agt.2014.2049


D. Mandal and M. N. Mukherjee

The intent of the present article is to do a further study of nearly mertizable
spaces from an altogether new approach. The notion of pseudo-base was intro-
duced and studied in [7], and here, we define regular and uniform pseudo-bases,
and ultimately achieve analogues of two well known results on metrizability in
our setting.

At the outset we recall a few definitions which may be found in [1, 2]. A
base B for a topological space X is called regular if for each x ∈ X and any
neighbourhood U of x, there exists a neighbourhood O of x such that the set of
all members of B that meet both O and X \ U, is finite; and a base B is called
a uniform base if for each x ∈ X and every neighbourhood U of x, the set of
all members of B that contain x and meet X \ U, is finite. It is clear that every
regular base is a uniform base. The next two metrization theorems are known
(see [1, 2]), which have been formulated in terms of the above special base.

Theorem 1.1.

(a). A T3-paracompact space X with a uniform base B is metrizable.
(b). Every T1-space X with a regular base B is metrizable.

As already proposed, our principal aim in this paper is to achieve analogous
versions of the results in Theorem 1.1 for near metrizability with accessories
formulated suitably.

In what follows, by a space X we shall mean a topological space X endowed
with a topology τ(say). The notations ‘clA’, ‘intA’ and ‘|A|’ will respectively
stand for the closure, interior and cardinality of a set A of a space X. A set
A(⊆ X) is called regular open if A = intclA, and the complement of a regular
open set is called regular closed. The set of all regular open (resp. closed) sets
of a space X will be denoted by RO(X)(resp. RC(X)). We shall sometimes
write A∗ for intclA for a subset A of X and C# = {A∗ : A ∈ C}, for any open
cover C of a space X.

Singal and Arya formulated the following definitions which are quite well
known by now.

Definition 1.2 ([10]). A topological space X is called nearly paracompact if
every regular open cover of X has a locally finite open refinement.

Definition 1.3 ([9]). A topological space X is said to be almost regular, if for
any regular closed set A and any x ∈ X \ A, there exist disjoint open sets U
and V in X such that x ∈ U and A ⊆ V .

2. Main results

We start by recalling a few definitions from [7] as follows:

Definition 2.1. If X and Y are two topological spaces, then a continuous,
injective map f : X → Y is called a pseudo-embedding of X into Y , if for any
A ∈ RO(X), f(A) is open.

If there is a pseudo-embedding f of X into Y , then we say that X is pseudo-
embeddable in Y . If a pseudo-embedding f : X → Y is surjective, we say that
f is a pseudo-embedding of X onto Y .

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Near metrizability

It is known [7] that every embedding is a pseudo-embedding; but the converse
is false.

Definition 2.2 ([7]). A space X is called nearly metrizable if it is pseudo-
embeddable in a metric space Y .

Definition 2.3 ([7]). Suppose B is a family of open subsets of X. We say that
B is a pseudo-base in X if for any A ∈ RO(X), there is a subfamily B0 of B
such that A =

⋃
{B : B ∈ B0}.

We now define a family B of open subsets of X to be a pseudo-base at a point
x ∈ X if for each U ∈ RO(X) containing x, there exists a B ∈ B such that
x ∈ B ⊆ U. Clearly, a family B of open subsets of a space X is pseudo-base
for X if and only if it is so at each x ∈ X.

We shall call a pseudo-base B σ-locally finite if B can be expressed as B =
∞⋃

n=1

Bn, where Bn is locally finite, for each n ∈ N.

We now define another type of bases as follows:

Definition 2.4. Let (X, τ) be a topological space.

(a) A family B of subsets of X is called nearly regular if for each U ∈ B
and any point x ∈ U, there exists a regular open set Ox containing x
such that the set {V ∈ B : V

⋂
Ox 6= φ and V

⋂
(X \ U) 6= φ} is finite.

(b) A pseudo-base B for X is called nearly regular if for each x ∈ X
and any regular open set Ox containing x, there exists a regular open
set Gx containing x such that the set {U ∈ B : U

⋂
Gx 6= φ and

U
⋂
(X \ Ox) 6= φ} is finite.

Remark 2.5. It is clear from the above definition that a subfamily of a nearly
regular family is a nearly regular family.

Proposition 2.6. If B is a nearly regular pseudo-base for a space X, then so
is B# = {B∗ : B ∈ B}.

Proof. First let x ∈ X and U a regular open set in X such that x ∈ U. As
B is a pseudo-base for X, there exists B ∈ B such that x ∈ B ⊆ U. Then
x ∈ B∗ ⊆ U∗ = U, and hence B# is a pseudo-base for X.
Next, let x ∈ X and Ox be any regular open set in X containing x. As B is
a nearly regular pseudo-base, there exists a regular open set Gx containing x
such that the set {B ∈ B : B

⋂
Gx 6= φ 6= B

⋂
(X \ Ox)} is finite.

It suffices to show that {B∗ ∈ B# : B∗
⋂
Gx 6= φ 6= B

∗
⋂
(X \ Ox)} is finite,

for which we need only to show that {B∗ ∈ B# : B∗
⋂
Gx 6= φ 6= B

∗
⋂
(X \

Ox)} ⊆ {B ∈ B : B
⋂
Gx 6= φ 6= B

⋂
(X \ Ox)}. In fact, B

⋂
Gx = φ ⇔

intclB
⋂
intclGx = φ ⇔ B

∗
⋂
Gx = φ, and B

⋂
(X \ Ox) = φ ⇒ B ⊆ Ox ⇒

B∗ ⊆ intclOx = Ox ⇒ B
∗
⋂
(X \ Ox) = φ. �

We shall call a space X to be an almost T3-space if it is almost regular and
Hausdorff.

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D. Mandal and M. N. Mukherjee

Theorem 2.7. A T2-space X, possessing a nearly regular pseudo-base B is an
almost T3-space.

Proof. Let F be a regular closed set and x ∈ X \F . Then there exists a regular
open set Ox containing x such that Ox

⋂
F = φ, i.e., F ⊆ X \ Ox.

By hypothesis, there exists a regular open set Gx containing x such that the
family U = {U ∈ B : U

⋂
Gx 6= φ and U

⋂
(X \ Ox) 6= φ} is finite. Put

O = Ox
⋂
Gx. Then O is a regular open set containing x such that O

⋂
F = φ.

Consider the family C = {U ∈ B : U
⋂
O 6= φ and U

⋂
F 6= φ}. Since

F ⊆ X \ Ox, C is finite.
Now for each U ∈ C, |U| ≥ 2 as O

⋂
F = φ.

Let B′ = B \ C. We show that B′ is a pseudo-base for X. In fact, let p ∈ X and
W a regular open set containing p. Let us enumerate C as {W1, W2, ..., Wn}
and let x1, x2,..., xn be points from W1, W2,..., Wn respectively different from
p.
Since X is T2, each {xi} is regular closed and so X \{x1, x2, ..., xn} is a regular
open set containing p and hence there exists a B1 ∈ B such that p ∈ B1 ⊆
X \ {x1, x2, ..., xn}. Again there exists B2 ∈ B such that p ∈ B2 ⊆ W . Thus
there exists B3 ∈ B such that p ∈ B3 ⊆ B1

⋂
B2 ⊆ W i.e., p ∈ B3 ⊆ W where

B3 6∈ C. This shows that B
′ is a pseudo-base for X.

Put G = {U ∈ B′ : U
⋂
F 6= φ} and G =

⋃
{U : U ∈ G}. Then F ⊆ G

and G
⋂
O = φ with x ∈ O (since for U ∈ G, if U

⋂
O 6= φ then U ∈ C, a

contradiction).
This shows that F and x are strongly separated. Thus X is almost regular and
consequently X is an almost T3-space. �

Definition 2.8 ([2]). Let X be a topological space and B a family of subsets
of X. An element U of B is called a maximal element of B if it is not contained
in any element of B other than U. We denote by m(B), the set of all maximal
elements of B and call m(B) the surface of B.

Theorem 2.9. Let B be a nearly regular family which is a cover of X. Then
the surface m(B) of B is a cover of X and is locally finite.

Proof. Let x ∈ X be taken arbitrarily and kept fixed, and let U ∈ B such that
x ∈ U. If U 6∈ m(B), then the family λU = {V ∈ B : V % U} is finite. In fact,
by definition of B, there exists a regular open set Ox containing x such that the
collection D = {V ∈ B : V

⋂
Ox 6= φ and V

⋂
(X \ U) 6= φ} is finite. Clearly,

λU ⊆ D and therefore λU is finite (note that x ∈ V
⋂
Ox). Consequently λU

has a maximal element V ′(say). Again x ∈ V ′ and V ′ ∈ m(B). Hence m(B) is
a cover of X.
We now show that m(B) is locally finite. As m(B) ⊆ B and B is nearly regular,
m(B) is nearly regular. Again every element of m(B) is maximal in m(B)
(because it is maximal in B and m(B) ⊆ B). Let x ∈ X. Then there exists a
U ∈ m(B) such that x ∈ U. Since m(B) is nearly regular, there exists a regular
open set Ox containing x such that the family B

′ = {V ∈ m(B) : V
⋂
Ox 6= φ

and V
⋂
(X \ U) 6= φ} is finite. But V \ U 6= φ for all V ∈ m(B) with V 6= U

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 28



Near metrizability

(because every element V in m(B) is maximal, there is no set L ∈ m(B) which
properly contains V ).
Thus {V ∈ m(B) : V

⋂
Ox 6= φ} = B

′
⋃
{U} is a finite set and hence m(B) is

locally finite. �

Theorem 2.10. A space possessing a nearly regular pseudo-base B is nearly
paracompact.

Proof. Let G be any regular open cover of X and let GB = {U ∈ B : ∃G ∈ G
with U ⊆ G}.
We check that GB is a pseudo-base for X. In fact, let x ∈ X and G be any
regular open set containing x. Now G being a cover, there exists G1 ∈ G such
that x ∈ G1. Thus G

⋂
G1 is a regular open set containing x. Since B is a

pseudo-base for X, there exists U ∈ B such that x ∈ U ⊆ G
⋂
G1 ⊆ G1 ∈ G ⇒

U ∈ GB with x ∈ U ⊆ G ⇒ GB is a pseudo-base for X.
Since B is nearly regular and GB ⊆ B, GB is nearly regular. Thus by Theorem
2.9, m(GB) is an open cover of X and locally finite. Also clearly m(GB) is an
open refinement of G. Hence X is nearly paracompact. �

Analogous to the concept of uniform base, we now define a special type of
base as follows:

Definition 2.11. A pseudo-base B for a space X is called a uniform pseudo-
base if for each x ∈ X and each regular open set Ox containing x, UOx = {U ∈
B : x ∈ U and U

⋂
(X \ Ox) 6= φ} is finite.

Lemma 2.12. Let B be a family of open sets of a space X such that B# is a
uniform pseudo-base for X. Then the surface m(B#) is a point finite regular
open cover of X.

Proof. Let x ∈ X. Then there exists U∗ ∈ B# (where U ∈ B) such that x ∈ U∗.
If U∗ 6∈ m(B#) then the set λU∗ = {V ∈ B

# : V ⊇ U∗} is finite. In fact, U∗

is a regular open set containing x and hence the family V = {V ∈ B# : x ∈ V
and V

⋂
(X \U#) 6= φ} is finite and λU∗ ⊆ V

⋃
{U∗}. Then λU∗ has a maximal

element m(λU# ) which is also a maximal element of B
# and which also contains

x. Hence m(B#) is a regular open cover of X.
We now show that m(B#) is point finite. If possible let x ∈ X be such that x
belongs to an infinite collection D of members of m(B#). Then we claim that
D is a pseudo-base for X at x.
If D is not a pseudo-base for X at x, there exists a regular open set W containing
x such that x ∈ D ⊆ W holds for no D ∈ D, i.e., for all D ∈ D, D

⋂
(X\W) 6= φ.

But {B ∈ D : B
⋂
(X \W) 6= φ} is finite as B# is a uniform pseudo-base. Hence

D is a pseudo-base for X at x.
Next let, U and V be two distinct (and hence non comparable) elements of D.
Since x ∈ U

⋂
V and U

⋂
V is a regular open set, there exists a W ∈ D such

that x ∈ W $ U
⋂
V (note that U

⋂
V 6∈ D, since otherwise U

⋂
V $ U would

contradict the maximality of U
⋂
V ), i.e., x ∈ W $ U and hence W is not a

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 29



D. Mandal and M. N. Mukherjee

maximal element of D although D ⊆ m(B#), a contradiction. Hence m(B#) is
a point finite regular open cover of X. �

Lemma 2.13. Let B be a family of open sets of a T2-space X such that B
#

is a uniform pseudo-base. Then there exists a countable family of point finite

regular open covers which taken together is a pseudo-base for X.

Proof. Let B
#
1 = B

# and B
#
2 = B

#
1 \ m

∗(B
#
1 ), where m

∗(B
#
1 ) is the collection

of all maximal elements of B
#
1 each of which contains at least two points. We

first show that B
#
2 is a pseudo-base for X. In fact, let x ∈ X. Then by Lemma

2.12, x belongs to only finitely many members U1, U2,..., Un (say) of m
∗(B

#
1 ).

Let xi ∈ Ui with x 6= xi for i = 1, 2, ..., n. Since X is T2, X \ {x1, x2, ..., xn} is
a regular open set containing x and so there exists B in B# such that x ∈ B ⊆
X \ {x1, x2, ..., xn}. Let W be any regular open set containing x. Then there
exists a B′ ∈ B# such that x ∈ B′ ⊆ W . Again there exists B1 ∈ B

# such that

x ∈ B1 ⊆ B
⋂
B′ ⇒ x ∈ B1 ⊆ W and B1 6∈ m

∗(B
#
1 ) [B1 ∈ m

∗(B
#
1 ) ⇒ B1 = Ui

for some i = 1, 2, ..., n ⇒ xi ∈ B1 but (xi 6∈ B) ⇒ B1 6⊆ B, a contradiction].

Therefore, x ∈ B1 ⊆ W and B1 ∈ B
#
2 . Again B

#
2 ⊆ B

#
1 and B

#
1 is a uniform

pseudo-base ⇒ B
#
2 is a uniform pseudo-base.

Now proceed by induction, if B
#
k

is already defined then put B
#
k+1

= B
#
k

\

m∗(B
#
k
) and as above, B

#
k+1

is a uniform pseudo-base for X. Then for each

n ∈ N, B#n is a uniform pseudo-base for X and so m(B
#
n ) is a point finite

regular open cover of X (by Lemma 2.12).
Consider an arbitrary x ∈ X. For each n ∈ N, choose Un ∈ m(B#n ) such that
x ∈ Un.
If there is n ∈ N satisfying |Un| = 1 then {Un : n ∈ N} is a pseudo-base at x.
If |Un| ≥ 2 for all n ∈ N then by definition of B#n , Un 6= Um for n 6= m. Hence
L = {Un : n ∈ N} is an infinite set of elements of the uniform pseudo-base
B#n , each containing x. We claim that L is a pseudo-base for X at x. If not,
then for some regular open set D containing x, there does not exist any C ∈ L
such that x ∈ C ⊆ D holds, i.e., for all C ∈ L, C

⋂
(X \ D) 6= φ. But since

L ⊆ B#, {V ∈ B# : x ∈ U and U
⋂
(X \ D) 6= φ} is finite, a contradiction.

Consequently, L is a pseudo-base for X at x. Hence {m(B#n ) : n ∈ N} is the
required family. �

Definition 2.14 ([11]). Let A be a family of subsets of a space X. The star
of a point x ∈ X in A, denoted by St(x, A), is defined by the union of all
members of A which contain x. A family A of subsets of a space X is said to
be a star refinement of another family B of subsets of X if the family of all
stars of points of X in A forms a covering of X which refines B.

Theorem 2.15 ([10]). An almost regular space X is nearly paracompact if and
only if every regular open covering of X has a regular open star refinement.

Definition 2.16. Let X be a topological space and Γ a family of covers of X.
We call Γ refined if for any point x ∈ X and any regular open set Ox containing

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 30



Near metrizability

x, there exists B ∈ Γ such that St(x, B) ⊆ Ox.
If all the members of Γ are regular open covers, then we say that Γ is a refined
family of regular open covers.

Theorem 2.17. Let B be a family of open sets of an almost T3 nearly para-
compact space X such that B# is a uniform pseudo-base for X. Then X has a
countable refined family of regular open covers.

Proof. By Lemma 2.13, there exists a countable family of point finite regular
open covers Bn, which taken together is a pseudo-base for X. Since X is almost
regular and nearly paracompact, by Theorem 2.15, each Bn has a regular open
star refinement Un.
Now fix x ∈ X, and for each n ∈ N, choose Bn ∈ Bn so that St(x, Un) ⊆ Bn.
Then {Bn : n = 1, 2, ...} is a pseudo-base for X at x. Let U be a regular open
set containing x. Then there exists Bk(say) such that x ∈ Bk ⊆ U and then
x ∈ St(x, Uk) ⊆ Bk ⊆ U. Thus {Un : n = 1, 2, ...} is a countable refined family
of regular open covers. �

Theorem 2.18 ([7]). A space X is nearly metrizable if and only if it is almost
T3 and possesses a σ-locally finite pseudo-base.

Theorem 2.19. Let X be an almost T3 nearly paracompact space such that

X has a countable refined family {Ui}
∞

i=1 of regular open covers. Then X is

nearly metrizable.

Proof. Since X is nearly paracompact, each Ui has a locally finite open refine-

ment Bi. Let B =

∞⋃

i=1

Bi. We show that B is a pseudo-base for X. In fact, let

x ∈ X and U be any regular open set containing x. Then since {Ui}
∞

i=1 is a
refined family of covers there exists k ∈ N such that x ∈ St(x, Uk) ⊆ U. But Bk
being a cover of X, there exists Bk ∈ Bk such that x ∈ Bk and Bk is contained
in some member of Uk containing x and hence is contained in St(x, Uk). Thus
x ∈ Bk ⊆ U. Hence B is a σ-locally finite pseudo-base for X and hence by
Theorem 2.18, X is nearly metrizable. �

Theorem 2.20. Let B be a family of open sets of an almost T3 nearly para-
compact space X such that B# is a uniform pseudo-base for X. Then X is
nearly metrizable.

Proof. Follows from Theorems 2.17 and 2.19. �

Theorem 2.21. Every almost T3-space X with a nearly regular pseudo-base B
is nearly mertizable.

Proof. By Theorem 2.10, X is nearly paracompact. Again by Proposition 2.6,
B# is a nearly regular pseudo-base. Since every nearly regular pseudo-base is a
uniform pseudo-base, B# is a uniform pseudo-base for X, and then by Theorem
2.20, it follows that X is nearly metrizable. �

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 31



D. Mandal and M. N. Mukherjee

Acknowledgements. The authors are grateful to the referee for some sug-

gestions towards certain improvement of the paper.

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c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 32