ShahDasAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 1, 2006

pp. 125-130

On nearly Hausdorff compactifications

Sejal Shah and T. K. Das

Abstract. We introduce and study here the notion of nearly Haus-

dorffness, a separation axiom, stronger than T 1 but weaker than T 2.

For a space X, from a subfamily of the family of nearly Hausdorff

spaces, we construct a compact nearly Hausdorff space rX containing

X as a densely C*-embedded subspace. Finally, we discuss when rX is

βX.

2000 AMS Classification: Primary 54C45, Secondary 54D35.

Keywords: Regular closed set, filter, compactification, Wallman base.

1. Introduction

A closed subset F in a topological space X is called a regular closed set if
F = Cl(IntF ). We denote the family of all regular closed subsets of X by
R(X). Observe that R(X) is closed under finite union. Also, if F ∈ R(X),
then Cl(X − F ) = X − IntF ∈ R(X). In Section 2, we define and study
the notion of a nearly Hausdorff space (nh-space). We introduce a topological
property Π and note that a space with property Π is an nh-space if and only if
it is Urysohn. A flow diagram showing various implications about separation
axioms supported by necessary counter examples is included in this section.
A map f :X→Y is called a density preserving map (dp-map) if for A ⊆ X,
Int(Clf (A)) 6= φ whenever IntA 6= φ [2]. We provide here an example showing
that the nh-property is not preserved even under continuous dp-maps. Note
that if X is an nh-space then R(X) forms a base for closed sets in X.

In Section 3, we obtain a ’βX like’ compactification of an nh-space X with
property Π. Since R(X) need not be closed under finite intersections, we form a
new collection Rf (X), of all possible finite intersections of members of R(X).
We observe that for an nh-space X with the property Π, the set rX = {α
⊆ Rf (X) | α is an r-ultrafilter} with the natural topology, is a nearly Haus-
dorff compact space which contains X as a dense C*-embedded subspace. The
natural question when rX = βX is discussed in Section 4. We observe that



126 S. Shah and T. K. Das

an nh-space X for which Rf (X) is a Wallman base, is a completely regular
Hausdorff space and hence for such a space X, rX = βX, the Stone-Čech com-
pactification of X. In particular, if X is normal or zero−dimensional then rX
= βX. The problem whether rX = βX for any Tychonoff space X is still open.

2. Nearly Hausdorff spaces

Definition 2.1. Distinct points x and y in a topological space X are said to
be separated by subsets A and B of X if x ∈ A−B and y ∈ B−A.
Definition 2.2. A topological space X is called nearly Hausdorff (nh-) if
for every pair of distinct points of X there exists a pair of regular closed sets
separating them.

Definition 2.3. A space X is said to have property Π if for every F ∈ R(X)
and x 6∈ F there exists an H ∈ R(X) such that x ∈ IntH and H ∩ F = φ. The
symbol X(Π) denotes a space X having property Π.

Remark 2.4. Henceforth all our regular spaces are Hausdorff. Recall that a
space X is Urysohn [5] if for each pair of distinct points of X, we can find dis-
joint regular closed sets of X containing the points in their respective interiors.
We have following implications:

Regular ⇒ Urysohn (Π) ⇔ Nearly Hausdorff (Π) ⇒ Urysohn
⇓

T1 ⇐ Nearly Hausdorff ⇐ Hausdorff
Examples given below (refer [4, 5]) justify that unidirectional implications

in the above flow diagram need not be revertible. In addition, Example 2.5(b)
shows that nearly Hausdorffness is not a closed hereditary property.

Example 2.5.

(a) An infinite cofinite space is a T 1 space but not an nh-space. The one-
point compactification of the space X in our Note 2 is a non-Hausdorff
compact nh-space.

(b) Consider N, the set of natural numbers with cofinite topology and I
= [0, 1] with the usual topology. Let X = N×I and define a topol-
ogy on X as follows: neighborhoods of (n, y), y 6= 0 will be usual
neighborhoods {(n, z) | y − ǫ < z < y + ǫ} in In = {n}×I for small
positive ǫ; neighborhoods of (n, 0) will have the form {(m, z) | m ∈ U ,
0 ≤ z < ǫm}, where U is a neighborhood of n in N and ǫm is a small
positive number for each m ∈ U . The resulting space X is a non
Hausdorff, nh-space without property Π. It is easy to observe that the
subspace N of X is closed but is a non-nh, T 1 space.

(c) Let A be the linearly ordered set {1, 2, 3, ...., ω, ...., -3, -2, -1} with
the interval topology and let N be the set of natural numbers with
the discrete topology. Define X to be A×N together with two distinct



On nearly Hausdorff compactifications 127

points say a and −a which are not in A×N. The topology ℑ on X
is determined by the product topology on A×N together with basic
neighborhoods M n(a) = {a} ∪ {(i, j) | i < ω, j > n} and M n(−a) =
{−a} ∪ {(i, j) | i > ω, j > n} about a and −a respectively. Resulting
space X is a non-Urysohn Hausdorff space without property Π. In fact,
there does not exist any regular closed set containing a and disjoint

from Mn(−a). This example also justifies that a Hausdorff space need
not have property Π.

(d) Let S be the set of rational lattice points in the interior of the unit
square except those whose x-coordinate is 1

2
. Define X to be S ∪

{(0, 0)} ∪ {(1, 0)} ∪ {( 1
2
, r

√
2) | r ∈ Q, 0 < r

√
2 < 1}. Topologize X

as follows: local basis for points in X from the interior of unit square
are same as those inherited from the Euclidean topology and for other
points following local bases are taken:
U n(0, 0) = {(x, y) ∈ S | 0 < x < 14 , 0 < y <

1
n
} ∪ {(0, 0)}, U n(1, 0)

= {(x, y) ∈ S | 3
4

< x < 1, 0 < y < 1
n
} ∪ {(1, 0)}, U n( 12 , r

√
2) =

{(x, y) ∈ S | 1
4
< x < 3

4
, |y−r

√
2| < 1

n
}.

The resulting space X is a Urysohn space without property Π.
(e) Let X be the set of real numbers with neighborhoods of non-zero points

as in the usual topology, while neighborhoods of 0 will have the form
U − A, where U is a neighborhood of 0 in the usual topology and A
= { 1

n
| n ∈ N}. Note that X is a non regular Urysohn space with

property Π.

Theorem 2.6. A nonempty product of an nh-space is an nh-space if and only
if each factor is an nh-space

Proof. Let {X γ}γ∈λ be a family of nh-spaces, λ 6= φ and let x, y ∈ X =∏
γ∈λ Xγ , x 6= y. Then xγ 6= yγ for some γ ∈ λ. Since each Xγ is an nh-space,

there exist regular closed sets Fx and Fy separating xγ and yγ . Define U =∏
β∈λ

Uβ and V =
∏

β∈λ
Vβ , where Vβ = Uβ = Xβ, for β 6= γ and Uγ = IntFx,

Vγ = IntFy. The regular closed sets ClU and ClV in X separate x and y.
Proof of the converse is similar. �

Lemma 2.7. Let X be an nh-space and let f :X→Y be a dp-epimorphism.
Then for a regular closed subset H of Y we have Clf (Clf −1(IntH)) = H and
hence R(Y ) = {Clf (F ) | F ∈ R(X)}.
Proof. Clearly for H ∈ R(Y ), Clf (Clf −1(IntH)) ⊆ H. For the reverse con-
tainment, if y ∈ H−Clf (Clf −1(IntH)) then there exists an open set U con-
taining y satisfying f −1(U∩IntH) = φ which contradicts y∈H = ClIntH. �
Note 1. Lemma 2.7 is stated in note 2.2 of [2] for a regular space. Further,
observe that the first projection of the space N×I in example 2.5 (b) shows that
continuous image of an nh-space need not be an nh-space. On the other hand,
if we consider second projection of N×I on [0, 1] with cofinite topology then we
get that even a continuous density preserving image of an nh-space need not be
an nh-space.



128 S. Shah and T. K. Das

3. The space rX

For an nh-space X, a filter α ⊆ Rf (X)−{φ} is called an r-filter. A maximal
r-filter is called an r-ultrafilter. The family of all r-ultrafilters in X is denoted
by rX. Observe that for x ∈ X, there exists a unique r-ultrafilter αx in
rX such that ∩αx = {x}. Further, if X is compact then each r-ultrafilter
in X is fixed. The converse is also true: If C is an open cover of X then
B = {F ∈ R(X)|X − U ⊂ F , for some U ∈ C} does not have finite intersection
property for otherwise B will generate a fixed r-ultrafilter which will contradict
that C is a cover of X. Hence C has a finite subcover. Topologize the set rX
by taking B = {F | F ∈ R(X)} as a base for closed sets in rX, where F =
{α ∈ rX | F ∈ α} and F ∈ R(X). The map r:X→rX defined by r(x) = αx,
where αx = {F ∈ Rf (X) | x ∈ F } is an embedding.

Lemma 3.1. Let X be an nh-space with property Π. Then the space rX of all
r-ultrafilters in X is a compact nh-space which contains X as a dense subspace.

Proof. Clearly αx = {F ∈ Rf (X)|x ∈ F } is an r-filter. For maximality of αx,
suppose A = ∩ni=1Ai in Rf (X) be such that A ∩ F 6= φ, for each F in αx.
If possible suppose for some i, Ai 6∈ αx. Then x 6∈ Ai. By the property Π,
there exists an H in R(X) such that x ∈ IntH and H ∩ Ai = φ. Therefore
H ∈ αx and hence H ∩ A 6= φ. But this implies φ 6= H ∩ A ⊂ H ∩ Ai = φ,
a contradiction. Further ClrX r(F ) = F for all F ∈ R(X) implies r is a dense
embedding. �

Note 2. A compactification of a non-Urysohn space without property Π may
also be an nh-space. For example, consider the subspace

Y = {( 1
n

,
1

m
) | n ∈ N, |m| ∈ N} ∪ {( 1

n
, 0) | n ∈ N}

of the usual space R2. Take X = Y ∪ {p+, p−}; p+, p− 6∈ Y and topologize it by
taking sets open in Y as open in X and a set U containing p+ (respectively p−)
to be open in X if for some r ∈ N, {( 1

n
, 1

m
) | n ≥r, m ∈ N} ⊆ U (respectively

{( 1
n
, 1

m
) | n ≥ r, −m ∈ N} ⊆ U ). The resulting space X is a non-Urysohn

Hausdorff space without property Π and its one point compactification is an
nh-space.

Proposition 3.2. Let the space X and rX be as in Lemma 3.1. Then X is
C*-embedded in rX.

Proof. Let f ∈ C*(X). Suppose range of f ⊆ [0, 1] = I. For α in rX, define
f ♯(α) = {H1 ∪ H2 ∈ R(I) | ClX f −1(IntH1 ∪ IntH2) ∈ α}. Note that if H1
∪ H2 ∈ f ♯(α) then either H1 ∈ f ♯(α) or H2 ∈ f ♯(α). Also f ♯(α) satisfies finite
intersection property. Thus ∩f ♯(α) 6= φ. We assert that ∩f ♯(α) = {t}, for
some t ∈ I.

Assuming the assertion in hand, we define rf : rX→I by rf (α) = ∩f ♯(α).
Clearly rf restricted to X is f . We now establish continuity of rf . Let
α ∈ rX. Then choose an open set G of I such that t ∈ G, where rf (α)=t. Using



On nearly Hausdorff compactifications 129

regularity of I successively we obtain open sets G1, G2 such that
t ∈ G1 ⊆ ClG1 ⊆ G2 ⊆ ClG2 ⊆ G. Set Ft = ClG2 and Ht = Cl(I − ClG1).
Since IntFt ∪ IntHt = I. We have Ft ∪ Ht ∈ f ♯(α) and as t 6∈ Ht, Ft ∈
f ♯(α) and Ht 6∈ f ♯ (α). If Lt = ClX f −1(IntHt), then α 6∈ Lt and the open set
rX−Lt contains α. Finally the containment rf (rX− Lt) ⊆ G establishes the
continuity. For the assertion, one may use the above technique to note that
{F ∈ R(I) | t ∈ IntF } ⊆ f ♯ (α), for each t ∈ f ♯(α). �

Theorem 3.3. Let X be an nh-space with property Π. Then there exists a
compact nh-space rX in which X is densely C*-embedded.

Proof. Follows from Lemma 3.1 and Proposition 3.2. �

Corollary 3.4. If X is a regular space, then it is densely C∗-embedded in rX.

4. When rX = βX?

Let X be an nh-space such that Rf (X) is a Wallman base. Then by 19L(7)
in [5], X is a completely regular space. Therefore by Corollary 3.4, X is C*-
embedded in rX. Further if X is an nh-space such that Rf (X) forms a Wallman
base then by 19L(5) in [5], rX is Hausdorff. Hence we have the following result:

Theorem 4.1. Let X be an nh-space such that Rf (X) is a Wallman base.
Then rX = βX.

Corollary 4.2. If X is normal or zero-dimensional then rX = βX.

Question: Is rX = βX when X is a Tychonoff space?

Acknowledgements. We thank the referee for his/her valuable sugges-
tions.

References

[1] E. Čech, Topological Spaces, (John Wiley and Sons Ltd., 1966).
[2] T. K. Das, On Projective Lift and Orbit Space, Bull. Austral. Math. Soc. 50 (1994),

445-449.
[3] J. R. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, (Springer-

Verlag, 1988).
[4] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, (Springer-Verlag,

1978).
[5] S. Willard, General Topology, (Addition-Wesley Pub. Comp., 1970).



130 S. Shah and T. K. Das

Received November 2004

Accepted July 2005

Sejal Shah

Department of Mathematics, Faculty of Science, The M. S. University of Bar-
oda, Vadodara, India.

T. K. Das (tarunkd@yahoo.com)
Department of Mathematics, Faculty of Science, The M. S. University of Bar-
oda, Vadodara, India.