EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 1260-1273
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

CC-Tychonoffness, CCT3 and CC-Almost Regularity

Sadeq Ali Thabit1, Wafa Alqurashi2,∗

1 Department of Mathematics, Faculty of Education-Almahra, Hadhramout University,
Yemen
2 Department of Mathematical Sciences, Faculty of Applied Sciences,
Umm Al-Qura University, Saudi Arabia

Abstract. Following the notion of so-called C-normality - a weaker version of normality in
topological spaces as proposed by A. V. Arhangel’skii, further weaker version called CC-normality
is studied by Kalantan et al [14]. In this paper, we investigate various type of properties such
as CC-complete regularity, CC-almost complete regularity, CC-regularity, CC-almost regularity,
CCT3 and CC-Tychonoffness. A space (X, T ) is called a CC-completely regular (resp. CC-almost
completely regular, CC-regular, CC-almost regular, CCT3, CC-Tychonoff) space if there exist a
completely regular (resp. almost completely regular, regular, almost regular, T3, Tychonoff) space
Y and a bijective function f : X → Y such that the restriction function f |A : A → f(A) is a
homeomorphism for each countably compact subspace A ⊆ X. We study these properties and
present some examples to illustrate the relationships among them with other forms of topological
properties.

2020 Mathematics Subject Classifications: 54C10, 54D10, 54D20, 54D15,54D70

Key Words and Phrases: C-normal, CC-normal, C-regular, C-Tychonoff, L-normal, L-regular
and L-Tychonoff

1. Introduction

The notion of C-normality has been studied by Alzahrani and Kalantan in [7]. The
notion of L-normality has been studied by Kalantan and Saeed in [12]. Then, Alzahrani
studied the notions of C-regularity, L-regularity, C-Tychonoff and L-Tychonoff in [5, 6].
At the end of 2022, Al-Awadi and others studied the notions of C-mild normality and
C-κ-normality [1]. Thabit studied the notion of epi-partial normality in [26]. At the
end of 2021, Thabit and others studied the notion of epi-quasi normality in [25]. Thabit
and Alqurashi studied the notions of C-almost normality and L-almost normality in [3].
Thabit and others studied the notions of C-complete regularity and CT3 and C-almost
regularity in [24]. The notions of LT3, L-complete regularity and L-almost regularity

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4776

Email addresses: sthabit1975@gmail.com, s.thabit@hu.edu.ye (Sadeq Ali Thabit),
wafa-math@hotmail.com, wkqurashi@uqu.edu.sa (Wafa Alqurashi)

https://www.ejpam.com 1260 © 2023 EJPAM All rights reserved.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1261

have been studied in [2]. The notion of CC-normality have been studied by Kalantan
and others in [14]. The notions of C,C2-paracompactness are studied in [19] and the
notions of L,L2-paracompactness are studied in [13]. In this paper, we investigate
the properties, CC-complete regularity, CC-regularity, CC-almost regularity, CC-almost
complete regularity, CCT3 and CC-Tychonoffness. We present some examples to illustrate
the relationships among these properties with other kinds of normality, complete regularity
and regularity. We need to recall that: a subset A of a space X is said to be a closed
domain subset if A = int(A) [15]. A subset A of a space X is called π-closed if it is a finite
intersection of closed domain subsets [27]. Two subsets A and B of a space X are said
to be separated if there exist two disjoint open subsets U and V of X such that A ⊆ U
and B ⊆ V [9, 10, 17]. If T and T ′ are two topologies on X such that T ′ ⊆ T , then
T ′ is called a topology coarser than T , and T is called finer [10]. A T4-space is a T1

normal space, a T3-space is a T1 regular space and a Tychonoff space is a T1 completely
regular space. A space X is said to be Hausdorff or a T2-space, if for each distinct two
points x, y ∈ X there exist two open subsets U and V of X such that x ∈ U , y ∈ V and
U ∩ V = ∅ [10]. A space X is said to be completely Hausdorff or Urysohn [10, 23], if for
each distinct two points x, y ∈ X there exist two open subsets U and V of X such that
x ∈ U , y ∈ V and U ∩V = ∅. A space X is said to be almost completely-regular if for each
x ∈ X and each closed domain subset F of X such that x ̸∈ F , there exists a continuous
function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1} [21]. A space X is said to be
almost-regular if for each x ∈ X and each closed domain subset F of X such that x ̸∈ F ,
there exist two disjoint open subsets U and V such that x ∈ U and F ⊆ V [20]. A space
X is said to be sub-metrizable [11], if there exists a metric d on X such that the topology
Td on X generated by d is coarser than T . The topology on X generated by the family
of all open domain subsets of X, denoted by Ts, is coarser than T , and (X, Ts) is called
the semi-regularization of X and the space (X, T ) is called semi-regular if T = Ts [16]. A
space X is called CC-normal [14] if there exist a normal space Y and a bijective function
f : X → Y such that the restriction function f |A : A → f(A) is a homeomorphism for
each countably compact subspace A ⊆ X. The basic definitions and any undefined terms
in this article can be found in [25] and [26].

2. Preliminaries

First, we present the main definitions of this work.

Definition 1. Let X be a space, then:

(1) A space X is called a CC-regular (resp. CC-almost regular) space if there exist a
regular (resp. almost regular) space Y and a bijective function f : X → Y such
that the restriction function f |A : A → f(A) is a homeomorphism for each countably
compact subspace A ⊆ X.

(2) A space X is called a CC-completely regular (resp. CC-almost completely regular)
space if there exist a completely regular (resp. almost completely regular) space Y



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1262

and a bijective function f : X → Y such that the restriction function f |A : A → f(A)
is a homeomorphism for each countably compact subspace A ⊆ X.

(3) A space X is called a CC-Tychonoff (resp. CCT3) space if there exist a Tychonoff
(resp. T3) space Y and a bijective function f : X → Y such that the restriction
function f |A : A → f(A) is a homeomorphism for each countably compact subspace
A ⊆ X.

From Definition 1, clearly that: every completely regular (resp. regular, almost
completely regular, almost regular, T3, Tychonoff) space is CC-completely regular (resp.
CC-regular, CC-almost completely regular, CC-almost regular, CCT3, CC-Tychonoff),
just by taking X = Y and the identity function, but the converses need not be true. The
next example is of a CC-Tychonoff, CCT3, CC-completely regular and CC-regular space
which is neither Tychonoff, T3, completely regular nor regular.

Example 1. The Smirnov’s deleted sequence topology: [23, Example 64], is a Urysohn,
Lindelöf first countable separable space which is not paracompact [23]. Since X is
a sub-metrizable space, by Corollary 1 and Theorem 1 we get: X is CC-Tychonoff,
CCT3, CC-completely regular, CC-regular, CC-almost regular and CC-almost completely
regular, but it is neither almost normal, Tychonoff, completely regular, T3 nor
regular. Also, the half disc topology [23, Example 78], is a CC-normal, CC-regular,
CC-completely regular, CC-Tychonoff, CCT3 and CC-almost completely regular space
being sub-metrizable, but it is neither regular, normal, completely regular, T3 nor
Tychonoff.

The following examples are CC-almost regular and CC-almost completely regular
spaces which are neither almost regular, L-almost regular nor almost completely regular:

Example 2. The relatively prime integer topology [23, Example 60], is a Hausdorff, semi
regular, Lindelöf, first countable separable space that is neither Urysohn, quasi normal,
almost regular nor regular [25, Example 2.9]. The spaceX is epi-mildly normal space which
is neither epi-quasi normal, epi-regular nor epi-completely regular [4, 25]. Since the space
X is Lindelöf non Urysohn, we conclude: it is neither C-normal, C-regular, C-completely
regular nor C-Tychonoff [24]. Thus, it is neither L-almost regular nor L-almost completely
regular [2]. By Theorem 2, it is neither CC-regular, CC-completely regular, CCT3,
CC-Tychonoff nor CC-normal. Since the space X is a Hausdorff first countable space,
by Theorem 17 and Corollary 10 we obtain that: the space X is CC-almost regular
and CC-almost completely regular. Observe that: any Hausdorff first countable Lindelöf
space is not necessary to be CC-regular, CCT3, CC-normal, CC-Tychonoff, epi-regular
nor Urysohn. This example also shows that CC-almost regularity does not imply L-almost
regularity.

Now, we present the following basic results.

Theorem 1. Every epi-completely regular space is CC-Tychonoff.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1263

Proof. By assumption, there exist a topology T ′ on X coarser than T such that
(X, T ′) is Tychonoff [4]. Thus, the identity mapping IX : (X, T ) → (X, T ′) is a bijective
continuous function. Let M be any countably compact subspace of (X, T ). Since a
continuous image of a countably compact subset is countably compact [10], we get: IX(M)
is a countably compact subspace of (X, T ′) as IX(M) = M is countably compact subspace
of both (X, T ) and (X, T ′). Thus, the restriction of the identity function (IX)|M : M →
Ix(M) is bijective continuous. Let U be any open set in (M, TM ). Since M is a countably
compact subset of (X, T ′), there exists an open set V in (X, T ′) and hence in (X, T ) such
that U = V ∩M . Thus, (IX)|M (U) = (IX)|M (V ∩M) = V ∩M = U , which is an open
set in (IX(M), T ′

M ). Hence, (IX)|M is open and hence a homeomorphism. Therefore, X
is CC-Tychonoff.

Note that: every epi-normal space is epi-almost normal and epi-almost normal space is
epi-completely regular [4]. Obviously, every epi-regular space is CCT3, every epi-normal
space is CC-normal and every epi-regular space is CC-regular.

Corollary 1. Every sub-metrizable (resp. epi-almost normal, epi-normal) space is
CC-Tychonoff.

The converses of Theorem 1 and Corollary 1 are not true in general as shown by the
next example:

Example 3. The countable complement topology: [23, Example 20], (R, CC) is a
T1-Lindelöf C-regular space, which is neither Hausdorff, regular, normal, first countable,
separable, paracompact nor L-regular [5, 23]. Also, (R, CC) is a CC-normal space, which
is not L-normal [14]. Since X is not Hausdorff, it is neither epi-regular, epi-normal nor
epi-mildly normal. Since the only countably compact subsets in X are finite subsets,
by Theorem 4 and Corollary 2 (R, CC) is CC-Tychonoff, CCT3, CC-completely regular
and CC-regular. This example shows that: CC-complete regularity, CC-normality, CCT3

and CC-Tychonoffness do not imply epi-regularity (resp. epi-complete regularity, epi-mild
normality, sub-metrizable, L-regularity, LT3, L-normality nor Hausdorffness). Also, it is
a CC-Tychonoff space which is not L-regular.

Theorem 2. Every CC-completely regular space is C-completely regular.

Proof. By assumption, there exist a completely regular space Y and a bijective function
f : X → Y such that the restriction function f |A : A → f(A) is a homeomorphism for
each countably compact subsets A ⊆ X. Let C be any compact subset of X. Since every
compact space is countably compact [10], we have: C is countably compact subset of X.
Thus, the restriction function f |C : C → f(C) is a homeomorphism. Since C was arbitrary
compact subset of X, we conclude that: X is C-completely regular.

Similarly, it is easy to prove that: every CC-regular space is C-regular, every
CCT3-space is CT3, every CC-Tychonoff space is C-Tychonoff, every CC-almost regular
space is C-almost regular and every CC-almost completely regular space is C-almost
completely regular.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1264

Theorem 3. Every CC-completely regular space is CC-almost completely regular.

Proof. By assumption, there exist a completely regular space Y and a bijective function
f : X → Y such that the restriction function f |A : A → f(A) is a homeomorphism for
each countably compact subsets A ⊆ X. Since every completely regular space is almost
completely regular [21], we obtain: Y is an almost completely regular space. Therefore,
X is CC-almost completely regular.

Similarly, every CC-completely regular space is CC-regular, every CC-regular space
is CC-almost regular, every CC-almost completely regular space is CC-almost regular,
every CC-Tychonoff space is CC-completely regular, every CCT3-space is CC-regular and
every CC-Tychonoff space is CCT3. The converses of Theorem 3 and stated facts are not
true in general. Here is an example of a CC-normal and CC-almost completely regular
space, which is neither CC-completely regular, CCT3, CC-Tychonoff nor CC-regular.

Example 4. The left ray topology (R,L) is a normal second countable and almost
completely regular space [23]. Therefore, (R,L) is a CC-normal and CC-almost completely
regular space, which is neither CCT3, CC-regular, CC-Tychonoff nor CC-completely
regular because it is not C-regular [5].

The next example is of a CC-completely regular space which is neither CCT3 nor
CC-Tychonoff.

Example 5. The odd-even topology [23, Example 6], is a regular, completely regular,
normal, locally compact, paracompact, separable, second countable space, which is neither
T0, compact, countably compact nor semi regular [23]. So, the odd-even topology is
a CC-regular, CC-completely regular, CC-normal and CC-almost completely regular
space, which is neither epi-regular nor epi-mildly normal. Observe that: the odd even
topology is neither CT3 nor LT3 [2, 24]. Hence, it is neither C-Tychonoff nor L-Tychonoff.
Therefore, it is neither CC-Tychonoff nor CCT3. Therefore, the odd-even topology is
a CC-completely regular and CC-normal space, which is neither CC-Tychonoff, CCT3

nor epi-regular. Note that: the odd even topology is C-paracompact space which is not
CC-regular.

Note that: CC-regularity does not imply CC-complete regularity, CCT3 does not
imply CC-Tychonoff and CC-almost regularity does not imply CC-almost complete
regularity. Here is a counterexample:

Example 6. The Tychonoff corkscrew topology: [23, Example 90], is a T3, regular, semi
regular and countably compact space, which is neither completely regular, normal, locally
compact, Lindelöf, second countable nor paracompact [23]. Since X is a T3-space, it is
epi-regular, CCT3 and CC-regular. Since X is countably compact space which is neither
almost completely regular nor epi-completely regular [4], we conclude that: it is neither
CC-completely regular, CC-Tychonoff nor CC-almost completely regular. Therefore, the
Tychonoff corkscrew topology is a CC-regular, CCT3 and CC-almost regular space, which
is neither CC-completely regular, CC-Tychonoff nor CC-almost completely regular.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1265

Observed that: any uncountable indiscrete space X is a CC-normal, CC-regular,
CC-completely regular and CC-almost completely regular space which is neither
epi-regular, CC-Tychonoff nor CCT3. The following example is a CC-almost completely
regular space, which is neither CCT3, CC-normal nor CC-regular.

Example 7. The particular point topology: [23, Example 10], (R, Tp) is a separable
first countable space which is neither Hausdorff, paracompact, regular nor normal [23].
(R, Tp) is neither a C-regular nor C-normal space [5, 7]. Then, it is neither CC-regular
nor CC-normal. Since the particular point topology (R, Tp) is an almost completely
regular space, it is both CC-almost regular and CC-almost completely regular. Therefore,
(R, Tp) is a CC-almost regular and CC-almost completely regular space, which is neither
CC-regular, CCT3, CC-completely regular, CC-normal, CC-Tychonoff nor epi-regular.

In view of the fact that: if X is a T1-space such that the only countably compact
subsets of X are the finite subsets, then X is CC-normal [14]. Then, we conclude:

Theorem 4. If X is a T1-space such that the only countably compact subsets of X are
the finite subsets, then X is CC-Tychonoff.

Proof. Let X be a T1-space. Let X = Y and consider Y with the discrete topology.
Then, the identity function IX : X → Y is a bijective function. If M is any countably
compact subspace of (X, T ), then by assumption M is a finite subspace of X and Y is with
a discrete topology. Since any finite countably compact subspace of a T1-space is discrete,
the restriction function (IX)|M : M → IX(M) = M is a homeomorphism because both
the domain and the co-domain are discrete, and they have the same cardinality. Since Y
is a Tychonoff space, we have: X is CC-Tychonoff.

Corollary 2. IfX is a T1-space such that the only countably compact subsets ofX are the
finite subsets, then X is CC-completely regular, CC-regular, CCT3, CC-almost regular
and CC-almost completely regular.

Theorem 5. If X is a countably compact CC-completely regular (resp. CC-Tychonoff,
CCT3, CC-regular, CC-almost completely regular, CC-almost regular) space, then X
is completely regular (resp. Tychonoff, T3, regular, almost completely regular, almost
regular).

Proof. Let X be a countably compact CC-completely regular (resp. CC-Tychonoff,
CCT3, CC-regular, CC-almost completely regular, CC-almost regular) space. Then, there
exist a completely regular (resp. Tychonoff, T3, regular, almost completely regular, almost
regular) space Y and a bijective function f : X → Y such that the restriction function
f |A : A → f(A) is a homeomorphism for each countably compact subspace A of X. Since
X is a countably compact space, put A = X. Since f is bijective, the function f : X → Y
is a homeomorphism. Since X ∼= Y , we get X is completely regular (resp. Tychonoff, T3,
regular, almost completely regular, almost regular).



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1266

Corollary 3. If X is a countably compact non-completely regular (resp. non-Tychonoff,
non T3, non-regular, non-almost completely regular, non-almost regular) space, then X
cannot be CC-completely regular (resp. CC-Tychonoff, CCT3, CC-regular, CC-almost
completely regular, CC-almost regular).

Recall that: a space X is called locally compact if X is Hausdorff and for each x ∈ X
and each open neighborhood V of x there exists an open neighborhood U of x such that
x ∈ U ⊆ U ⊆ V and U is compact [10]. In view of the fact that: every locally compact
space is Tychonoff [10], we get the following corollary:

Corollary 4. Every locally compact space is CC-Tychonoff.

Recall that: a space X is said to be mildly normal [22], if any pair of disjoint closed
domain subsets A and B of X can be separated. The converse of Corollary 4 is not true
in general as shown by the next example:

Example 8. The modified Dieudonné plank topology: [14, Example 2.4, Example 3.3],
is a Tychonoff, L-normal and CC-normal space, which is neither mildly normal nor
locally compact [14]. Thus, the modified Dieudonné plank is a CC-Tychonoff, CCT3,
CC-completely regular and CC-regular space, which is neither locally compact nor mildly
normal.

Note that: if X is a CC-almost completely regular (resp. CC-almost regular,
CC-complete regularity, CC-regular, CCT3, CC-Tychonoff) space and f : X → Y is a
witness of the CC-almost complete regularity (resp. CC-almost regularity, CC-complete
regularity, CC-regularity, CCT3, CC-Tychonoffness) of X, then f is not necessary to be
continuous. Here is a counterexample:

Example 9. Consider the countable complement topology on R, (R, CC). The only
countably compact subspaces are finite subspaces and (R, CC) is T1-space. Hence, (R, CC) is
CC-Tychonoff (hence CC-completely regular, CC-almost completely regular, CC-almost
regular, CCT3 and CC-regular). It is well known that the finite countably-compact
subspaces in a T1-space are discrete. If we let D be the discrete topology on R, then
the identity function from (R, CC) onto (R,D) is a witness of the CC-Tychonoffness (resp.
CC-complete regularity, CC-almost complete regularity, CC-almost regularity, CCT3,
CC-regularity) of (R, CC), which is not continuous.

Recall that: a space X is called a Fréchet if for any subset B of X and any x ∈ B,
there exists a sequence (an)n∈N of points of B such that an −→ x [10]. Thus, we conclude:

Theorem 6. If X is a CC-completely regular Fréchet space, then any function bears the
CC-complete regularity of X is continuous.

Proof. Similar to the proof of Theorem 2.9 in [14].

The proof of the next theorem is also similar to the proof of Theorem 2.9 in [14]..



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1267

Theorem 7. If X is a CC-Tychonoff (resp. CC-regular, CCT3, CC-almost
regular, CC-almost completely regular) Fréchet space, then any function bears the
CC-Tychonofness (resp. CC-regularity, CCT3, CC-almost regularity, CC-almost
complete regularity) of X is continuous.

Since every first countable space is Fréchet [10], we get the next corollary:

Corollary 5. If X is a CC-almost regular first countable space and f : X → Y is a
witness of the CC-almost regularity of X, then f is continuous.

Next, we introduce the following results:

Proposition 1. If X is a T1 CC-completely regular space, then the witness Y is Tychonoff.

Proof. LetX be a T1 CC-completely regular space. SinceX is a CC-completely regular
space, there exist a completely regular space Y and a bijective function f : (X, T ) →
(Y, T ′) such that f |A : A → f(A) is a homeomorphism for each countably compact subset
A ⊆ X. Suppose Y is not Tychonoff, then Y cannot be T1 because it is completely
regular. Then, there exist two distinct elements x and y in Y such that if U is any open
neighborhood of x, then y ∈ U or if V is any open neighborhood of y , then x ∈ V .
Thus, the set M = {f−1({x}), f−1({y})} is a T1 countably compact subspace of X. Then,
f |M : M → f(M) is a homeomorphism. But f(M) = {x, y} cannot be T1, which is a
contradiction. Hence, Y must be T1 and thus Tychonoff.

Similarly, we can prove the next proposition:

Proposition 2. If X is a T1 CC-regular (resp. CC-normal) space, then the witness Y is
T3 (resp. T4).

Thus, we get the next corollary:

Corollary 6. Every T1 CC-completely regular (resp. CC-regular, CC-normal) space is
CC-Tychonoff (resp. CCT3, CC-Tychonoff).

Theorem 8. Every T1 CC-completely regular Fréchet (resp. first countable) is
epi-completely regular.

Proof. Let X be a T1 CC-completely regular Fréchet space (resp. first countable).
Then, there exist a completely regular space Y and a bijective function f : (X, T ) →
(Y, T ′) such that f |A : A → f(A) is a homeomorphism for each countably compact subset
A ⊆ X. Since X is Fréchet (resp. first countable), we have f is continuous. Since X
is T1 CC-completely regular, by Proposition 1 we obtain Y is Tychonoff. Now, define
a topology T ⋆ on X as follows: T ⋆ = {f−1(U) : U ∈ T ′}. Clearly, T ⋆ is a topology
on X coarser than T such that f : (X, T ⋆) → (Y, T ′) is continuous. If W ∈ T ⋆, then
W = f−1(U) for some open set U in T ′. So, f(W ) = f(f−1(U)) = U , which is open set
in (Y, T ′). Thus, f : (X, T ⋆) → (Y, T ′) is open and hence a homoeomorphism. Therefore,
(X, T ⋆) is a Tychonoff space. Since T ⋆ ⊆ T , we get: (X, T ) is epi-completely regular.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1268

Similarly, every T1 CC-regular Fréchet (resp. first countable) is epi-regular, every
CCT3-Fréchet (resp. first countable) is epi-regular, every CC-Tychonoff Fréchet (resp.
first countable) is epi-completely regular and every T1 CC-normal Fréchet (resp. first
countable) is epi-normal.

Corollary 7.

(1) Every CC-regular T1-first countable space is Urysohn.

(2) Every CCT3-first countable space is Urysohn.

By using Theorem 8 and Proposition 1, we can prove the next result as follows:

Theorem 9. Every T1 CC-regular Fréchet (first countable) Lindelöf space is epi-normal.

Proof. Let X be a CC-regular T1 Fréchet (resp. first countable) Lindelöf space. Then,
there exist a regular space Y and a bijective function f : (X, T ) → (Y, T ′) such that
f |A : A → f(A) is a homeomorphism for each countably compact subset A ⊆ X. Since X
is Fréchet (resp. first countable), we get f is continuous. Since the continuous image of
a Lindelöf space is Lindelöf [10], we obtain: Y is Lindelöf. Since Y is a regular Lindelöf
space, we have (Y, T ′) is normal. By Proposition 2, (Y, T ′) is a T3-space. Thus, (Y, T ′) is
a Hausdorff normal space and hence a T4-space. Define a topology T ⋆ on X as follows:
T ⋆ = {f−1(U) : U ∈ T ′}. By using the same arguments to the proof of Theorem 8, we
obtain: f : (X, T ⋆) → (Y, T ′) is a homoeomorphism. Since (Y, T ′) is a T4-space, we have:
(X, T ⋆) is T4. Since T ⋆ ⊆ T , we get: (X, T ) is epi-normal.

Corollary 8.

(1) Every T1 CC-completely regular Fréchet (first countable) Lindelöf space is epi-normal.

(2) Every CCT3-Fréchet (first countable) Lindelöf space is epi-normal.

Theorem 10. Every CC-regular Fréchet (resp. first countable) Lindelöf space is
CC-normal.

Proof. It is similar to that of Theorem 9.

Corollary 9. Every CC-completely regular (resp. CC-Tychonoff, CCT3) first countable
Lindelöf space is CC-normal.

It is obvious that every CC-completely regular (resp. CC-regular, CCT3,
CC-Tychonoff) countably compact Lindelöf space is CC-normal. The proof of the next
results is similar to that of Theorem 3.5 in [14]:

Theorem 11. If X is a CC-completely regular (resp. CC-regular, CC-Tychonoff,
CCT3) space, then the Alexandroff duplicate A(X) of X is CC-completely regular (resp.
CC-regular, CC-Tychonoff, CCT3).



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1269

3. Some properties and relationships

Now, we present the next results: the proof of the next theorem is similar to that of
Theorem 2.7 in [14].

Theorem 12. CC-Tychonoffness, CCT3, CC-complete regularity, CC-regularity,
CC-almost regularity and CC-almost complete regularity are topological properties.

The proof of the following results is similar to the proof of Theorem 2.8 in [14]:

Theorem 13. CC-Tychonoffness, CCT3, CC-complete regularity, CC-regularity,
CC-almost regularity and CC-almost complete regularity are additive properties.

Theorem 14. CC-complete regularity, CC-Tychonoffness, CCT3 and CC-regularity are
hereditary properties.

Proof. Let X be a CC-completely regular (resp. CC-Tychonoff, CCT3, CC-regular)
space. Pick a completely regular (resp. Tychonoff, T3, regular) space Y and a bijective
function f : X → Y such that f |A : A → f(A) is a homoeomorphism for each countably
compact subspace A ⊆ X. Let M be any subspace of X and let N = f(M) ⊆ Y .
Then, N is a completely regular (resp. Tychonoff, T3, regular) space because it is a
subspace of a completely regular (resp. Tychonoff, T3, regular) space Y . Now, we have:
f |M : M → f(M) is a bijective function. Since any countably compact subspace K
of M is countably compact subset in X and (f |M )|K = f |K , we conclude that: M is
CC-completely regular (resp. CC-Tychonoff, CCT3, CC-regular).

Theorem 15. If X is an L-Tychonoff space such that each countably compact subspace
is contained in a Lindelöf subspace, then X is CC-Tychonoff.

Proof. Let X be an L-Tychonoff space such that if A is a countably compact subspace
ofX, there exists a Lindelöf subspace B ofX such that A ⊆ B. Let Y be a Tychonoff space
and f : X → Y be a bijective function such that f |C : C → f(C) is a homeomorphism for
each Lindelöf subspace C ⊆ X. Now, let A be a countably compact subspace of X. Pick a
Lindelöf subspace B of X such that A ⊆ B. Then, f |B : B → f(B) is a homeomorphism.
Thus, f |A : A → f(A) is a homeomorphism as (f |B)|A = f |A. Hence, X is CC-Tychonoff.

We can find some statements analogous to that of Theorem 15. Here are some of them:

Theorem 16.

(1) If X is a C-Tychonoff (resp. C-completely regular, CT3, C-regular, C-almost regular,
C-almost completely regular) space such that each countably compact subspace is
contained in a compact subspace, then X is CC-Tychonoff (resp. CC-completely
regular, CCT3, CC-regular, CC-almost regular, CC-almost completely regular).

(2) If X is a CC-Tychonoff (resp. CC-completely regular, CCT3, CC-regular, CC-almost
regular, CC-almost completely regular) space such that each Lindelöf subspace is
contained in a countably compact subspace, then X is L-Tychonoff (resp. L-completely
regular, LT3, L-regular, L-almost regular, L-almost completely regular).



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1270

(3) If X is an L-completely regular (resp. L-completely regular, LT3, L-regular,
L-almost regular, L-almost completely regular) space such that each countably compact
subspace is contained in a Lindelöf subspace, then X is CC-completely regular (resp.
CC-completely regular, CCT3, CC-regular, CC-almost regular, CC-almost completely
regular).

Theorem 17. If X is a Hausdorff Fréchet (resp. first countable) space, then X is
CC-almost completely regular.

Proof. Let X be a Hausdorff Fréchet (resp. first countable) space. Then, there
exists a topology T ′ coarser than T such that (X, T ′) is T1-almost completely regular
[4]. The identity function IX : (X, T ) → (X, T ′) is a bijective continuous function. Let
M be any countably compact subspace of (X, T ). Then, (IX)|M : M → IX(M) is a
bijective continuous function and IX(M) = M is a countably compact subset in both
(X, T ) and (X, T ′). Let U be any open set in (M, TM ). Since M is a countably compact
subset of (X, T ′), there exists an open set V in (X, T ′) such that U = V ∩ M . Thus,
(IX)|M (U) = (IX)|M (V ∩M) = V ∩M = U , which is an open set in (IX(M), T ′

M ). Hence,
(IX)|M is open and hence a homeomorphism. Therefore, X is CC-almost completely
regular.

Corollary 10. If X is a Hausdorff Fréchet (resp. first countable) space, then X is
CC-almost completely regular.

Theorem 18. Every Hausdorff almost completely regular space is CC-Tychonoff.

Proof. Let (X, T ) be a Hausdorff almost completely regular space. Let (X, Ts) be
the semi-regularization of (X, T ). Then, (X, Ts) is a Hausdorff completely regular space
because a semi-regularization of a Hausdorff almost completely regular space is Hausdorff
completely regular [16]. Hence, (X, Ts) is Tychonoff. Since Ts ⊆ T , we obtain that (X, T )
is epi-completely regular. By Theorem 1, we conclude that (X, T ) is CC-Tychonoff.

Similarly, we can prove the next result:

Theorem 19. Every Hausdorff almost regular space is a CCT3-space.

Observed that: CC-normality does not imply CC-almost regularity. Here is a
counterexample.

Example 10. The excluded point topology: [23, Example 15], (X, Ep) is a T0, compact,
paracompact, first countable and normal space, which is neither T1, regular, separable nor
semi regular [23]. (X, Ep) is not almost regular [24]. Hence, it is not almost completely
regular. Since X is a countably compact space which is not almost regular, by Corollary
3, we obtain: X is neither CC-almost regular, CC-regular, CC-completely regular,
CC-almost completely regular, CCT3 nor CC-Tychonoff. Since X is a normal space,
it is CC-normal. Since X is not T1, we obtain: X is neither epi-regular nor epi-mildly
normal. Therefore, the space (X, Ep) is a CC-normal space, which is neither CC-almost
regular, CC-regular, CC-Tychonoff nor CCT3.



S. A. Thabit, W. Alqurashi / Eur. J. Pure Appl. Math, 16 (2) (2023), 1260-1273 1271

Here is another example of a CC-Tychonoff space, which is not sub-metrizable.

Example 11. The deleted Tychonoff plank: [23, Example 87], is a Hausdorff and locally
compact space [23]. By Corollary 4, the deleted Tychonoff plank is a CC-Tychonoff,
CCT3, CC-completely regular and CC-regular space. Hence, it is CC-almost completely
regular and CC-almost regular. The deleted Tychonoff plank is also neither almost-normal
nor sub-metrizable [5, 7].

Any CC-completely regular (resp. CC-regular, CCT3, CC-Tychonoff) space is not
necessarily locally compact nor CC-normal as shown by the next example:

Example 12. Consider the Example 10 in [18], let G = Dω1 , where D = {0, 1} with the
discrete topology. Let H be a subspace of G consisting of all points of G with at most
countably many non zero coordinates. Put X = G × H. Raushan Buzyakova proved
that X cannot be mapped onto a normal space Y by a bijective continuous function
[8]. Observe that: H is T2-Fréchet and hence H is a k-space. The space G is also a
T2-compact space. Hence, X = G×H is a k-space [18]. Since X is Tychonoff, we get X is
CC-Tychonoff. Hence, it is a CC-completely regular, CCT3, CC-regular and CC-almost
completely regular space, which is not C-normal [18]. Since X is not C-normal, we obtain
X is neither CC-normal, sub-metrizable, C2-paracompact nor epi-normal. Note that:
every C2-paracompact space is C-normal [19]. The space X is also not locally compact.
Thus, the space X is a CCT3, CC-regular, CC-completely regular and CC-Tychonoff
space, which is neither CC-normal, C2-paracompact, epi-normal, sub-metrizable nor
locally compact.

Since every Hausdorff paracompact space is T4, the proof of the next result is similar
to that of Theorem 9:

Theorem 20. Every C2-paracompact first countable space is epi-normal (hence
epi-completely regular).

Thus, we obtain the next corollary:

Corollary 11. Every C2-paracompact first countable space is CC-Tychonoff.

Note that: the space presented in Example 2.25 in [19], is a C-paracompact first
countable space which is neither CC-regular nor L-almost regular because it is a Lindelöf
space that is neither almost regular nor C-regular. The following problems are still open
in this research.

Problems:

(1) Is there an example of a C-Tychonoff space which is not CC-almost regular?.

(2) Is there an example of a C2-paracompact space which is not CC-regular?.

(3) Is there an example of an L-Tychonoff space which is not CC-regular?.

(4) Are CC-complete regularity, CC-Tychonoffness, CCT3 and CC-regularity
multiplicative properties?.



REFERENCES 1272

4. Conclusion

New topological properties, called CC-complete regularity, CC-almost complete
regularity, CC-almost regularity, CCT3, CC-Tychonoffness and CC-regularity have been
studied in this work. Some results, properties, relationships and counterexamples have
been given and discussed.

Acknowledgements

The authors would like to thank the anonymous referee for his/her comments that will
help us improve this article.

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