@ Appl. Gen. Topol. 15, no. 1 (2014), 1-9doi:10.4995/agt.2014.2019
c© AGT, UPV, 2014

Hausdorff connectifications

Solai Ramkumar

Department of Mathematics, Alagappa University, Karaikudi - 630 003, Sivagangai District, Tamil

nadu, India. (ramkumarsolai@gmail.com)

Abstract

Disconnectedness in topological space is analyzed to obtain Hausdorff

connectifications of that topological space. Hausdorff connectifications

are obtained by some direct constructions and by some partitions of

connectifications. Also lattice structure is included in the collection of

all Hausdorff connectifications.

2010 MSC: 54D35; 54D05; 54D40.

Keywords: H-closed sets; cut points, n-disconnected set; pointly connected

mapping; Connectifications; Remainders of Connectifications;

Lattice isomorphism.

1. Introduction

An extension of a topological space X is a topological space that contains X
as a dense subspace. If an extension is a connected space, then that extension
is called a connectification of that space. There is a book of J. R. Porter and R.
Grandwoods that is devoted for Hausdorff extensions. This paper also studies
only Hausdorff spaces and Hausdorff connected extensions. It is easy to see that
if X has a proper compact (or H-closed) open subset, then X has no Hausdorff
connectification. There are spaces which can not have connectification. Porter
and Grandwoods gave some nice examples of Hausdorff spaces in [7], that can
not be densely embedded in a connected Hausdorff space. Several papers have
been devoted to connectifications (See: [1], [2], [3], [9]). Fedeli and Le Donne
in [4] proved that a T1-space can be densely embedded in a pathwise connected
T1-space if and only if it has no isolated points. Charatonik in [3] considered
generalized linear graphs to obtain Hausdorff connectifications and character-
ized the one point Hausdorff connectification of a subspace of a generalized

Received December 2011 – Accepted May 2013

http://dx.doi.org/10.4995/agt.2014.2019


S. Ramkumar

linear graph in [2]. Section 2 contains some basic ideas about disconnectedness
of topological spaces. section 3 presents some direct constructions to obtain
Hausdorff connectifications of a topological space. We also obtain Hausdorff
connectifications through remainders in section 4. Final section proves that if
f is a continuous and connected mapping from X onto Y such that f seper-
ates every pair of disjoint regular open subsets of X, then the lattice C(X) is
isomorphic to C(Y ). All spaces under consideration are Hausdorff topological
spaces.

2. Some Disconnected Spaces

Definition 2.1. A subset A of a space X is n-disconnected if A has exactly
n + 1 no. of clopen subsets in A except ∅ and A.

Definition 2.2. A subset A of a space X is countably infinite disconnected if
A has only countably infinite number of clopen subsets in A. A is countably
disconnected if it is either n-disconnected or countably infinite disconnected.

Definition 2.3. A subset A of a space X is uncountably disconnected if A is
not countably disconnected.

Example 2.4. (i) (0, 1) ∪(1, 2) is 1-disconnected.

(ii)
n
∪

k=1
(k, k + 1) is (2n − 2)-disconnected.

(iii) Set of all irrationals is uncountably disconnected subset of R.

Theorem 2.5. Let f : X(⊆ Z1) → Z be a continuous mapping such that
f(X) = Y ⊆ Z. If Y is n-disconnected subset of Z, then X is atleast a n-
disconnected subset of Z1. Also, if A is a component in Y , then f

−1(A) is a
component in X.

Proof. Let Y be a n-disconnected subset of a space Z. Then Y has n + 1 no of
clopen subsets. Let them be {A1, A2, · · · · · · An+1}. Since f is continuous, X
has atleast n+ 1 no of clopen subsets namely, {B1, B2, · · · · · · Bn+1}. Also, if A
is a component in Y , then f−1(A) is a component in X. If not, then there is a
connected subset C of X containing f−1(A). Then f(C) is a connected subset
of Y containing A, which is a contradiction to the maximality of A. �

Theorem 2.6. Let f : X → Z be an one to one and open mapping. If
Y = f(X) and if X is a n-disconnected subset of Z1, then Y is atleast a n-
disconnected subset of Z. Also, image of a component under the mapping f is
a component in Y .

Proof. If f is an one to one and open mapping, then f−1 : Y → X is a
continuous mapping. By the previous theorem 2.5, X is a n-disconnected
subset of Z1. �

A subspace of a n-disconnected space need not be a n-disconnected space.
Consider a subspace A = [0, 1] ∪[2, 3] of [0, 1] ∪[2, 3] ∪[4, 5], for some fixed n.
Then A is 1-disconnected subspace of a 2-disconnected space [0, 1] ∪[2, 3] ∪[4, 5].

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 2



Hausdorff connectifications

Theorem 2.7. Let A be subset of a n-disconnected space X. Then A is n-
disconnected if A is a connected dense subspace of X.

Definition 2.8. A point x ∈ Y is a cut point of X ⊆ Y if there is a clopen
subset A of X such that A ∪{x} ∪(X\A) is a connected subset of Y .

Theorem 2.9. If X(⊆ Y ) has n cut points, then X is atleast a n + 1-
disconnected subspace of Y .

Proof. If X has n cut points (say) p1, p2, · · · · · · pn, then there are n-clopen
subsets {Ai : i = 1, 2, · · · · · · n} such that Ai ∪(X\Ai) ∪{pi} is a connected
subset of Y . Thus X has 2n clopen subsets. Also unions and intersections of
clopen subsets are clopen subsets which increases the no of clopen subsets of
X. Thus X is atleast a n + 1-disconnected subspace of Y . �

There may be a clopen subset Ak of X such that Ak ∪(X\Ak) ∪ E is a
connected subset of Y , where E is a subset of Y and E contains more than one
point. This also increases the no of clopen subsets of X.

Theorem 2.10. Let X be a subspace of a space Y . If X is n-disconnected
subspace of Y , then X has atmost n cut points.

Theorem 2.11. A point x ∈ Y is a cut point of X ⊆ Y if and only if there is a
connected subset C of Y such that C ∩ X = A ∪(X\A) and Y \(C ∩ X) = {x},
where A is a clopen subset of X.

Proof. Proof follows directly from the definition of cut point. �

3. Some Connectifications

Theorem 3.1. Let X be a space having no isolated points. If X has finite
number of distinct clopen subsets(2n-disconnected space) such that each clopen
subset is neither H-closed nor compact, then there is an extension Y of X such
that Y is connected and hence Y is a connectification of X.

Proof. Let X be a space having 2n distinct clopen subsets of X. Let them
be {A1, A2, · · · · · · A2n}. Let A11 = A1. Since X\A1 is a clopen subset of
X, X\A1 is one of the member in {A2, · · · · · · A2n}. Let it be A12. Let A21 ∈
{A1, A2, · · · · · · A2n}\{A11, A12}. Then X\A21 ∈ {A1, A2, · · · · · · A2n}\{A11, A12, A21}.
Let it be A22. Thus we can arrange the clopen subsets {A1, A2, · · · · · · A2n} as
{A11, A12, A21, A22 · · · · · · An1, An2}. Let Y = X ∪{p1, p2, · · · · · · pn}, where
pi /∈ X for every i ∈ {1, 2, · · · · · · n}. Define the open neighborhoods of pi be
Ui1 ∪ Ui2 ∪ pi, where Uij ∈ Uij and Uij is a decreasing sequence of nonempty
open subsets of Aij’s. Then Y is an extension of X. If Y is not connected,
then there exists a clopen subset C of Y such that C ∩ X is a clopen subset
of X. By construction, there are some Ai1 and Ai2 such that C ∩ X = Ai1
or C ∩ X = Ai2. Then there is an element pi ∈ Y such that every open
neighborhood of pi intersects C ∩ X and X\C. Thus pi ∈ clY (C ∩ X) and
pi ∈ clY (X −C). That is pi ∈ clY (C) = C and pi ∈ clY (Y −C) = Y −C. That

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 3



S. Ramkumar

is pi ∈ C ∩(Y − C), which is a contradiction. It remains only to prove that Y
is a Hausdorff space. Let x, y ∈ Y and x 6= y.

Case: 1. Let x, y ∈ Y \X. Then there exists pi and pj such that x = pi
and y = pj. We can find nonempty open subsets Ui1, Ui2, Vj1 and Vj2 of
Ai1, Ai2, Aj1 and Aji2, respectively such that Ui1 ∪ Ui2 ∪{pi} and Vj1 ∪ Vj2 ∪{pj}
are open sets in Y containing x and y, respectively. Let x1 ∈ Ui1, x2 ∈ Ui2,
x3 ∈ Vj1 and x4 ∈ Vj2. Find disjoint open neighborhoods Wx1, Wx2, Wx3, Wx4
for x1, x2, x3, x4 such that Wx1 ⊆ Ui1, Wx2 ⊆ Ui2, Wx3 ⊆ Vj1 and Wx4 ⊆ Vj2.
Let U = Wx1 ∪ Wx2 ∪{pi} and V = Wx3 ∪ Wx4 ∪{pj}. Thus U and V are
disjoint open subsets of pi and pj, respectively.

Case: 2. Let x ∈ X and y ∈ Y \X. Then there exists a clopen subset Ai in
X and pi ∈ Y such that y = pi and x ∈ Ai1 or x ∈ Ai2. Let us assume that
x ∈ Ai1. Choose any open neighborhood U1x ∪ U2x ∪{pi} of pi and x0 ∈ U1x
such that x 6= x0. Then there are disjoint open sets Vx ⊆ U1x and Vx0 ⊆ U1x
such that x ∈ Vx and x0 ∈ Vx0. Let W = Vx0 ∪ Wx0 ∪{pi}, where Wx0 is any
nonempty open subset in Ai2. Then Ux and W are disjoint open neighborhoods
of x and y, respectively.

Case: 3. Let x, y ∈ X. Then there are disjoint open neighborhoods for x
and y in X. By our construction, They are also open in Y . �

If X is finitely disconnected and has no isolated points, then we can give
a one point connectification of X. we can attach only one point, whose open
neighborhoods are of the form U1 ∪ U2 ∪ · · · · · · Un, where Ui ∈ Ui and Ui is
decreasing sequence of non empty open subsets of the clopen subset Ai of X.

Corollary 3.2. If X is a space having countable number of distinct clopen sub-
sets of X(countably disconnected space) such that each clopen subset is neither
H-closed nor compact and having no isolated points, then there is an extension
Y of X such that Y is connected.

Corollary 3.3. If X is a space having uncountable number of distinct clopen
subsets of X(uncountably disconnected space) such that each clopen subset is
neither H-closed nor compact and having no isolated points, then there is an
extension Y of X such that Y is connected.

Definition 3.4. Let f : X → Y be a mapping from X to Y . Then f is said
to separates disjoint subsets A and B of X if f(A) ∩ f(B) = φ.

Theorem 3.5. Let f : X → Y be a continuous mapping from X onto Y such
that f seperates every pair of disjoint regular open subsets of X. If ξX is any
connectification of X, then there is a connectification ζY for Y .

Proof. Let ζY = Y ∪ K, where K = ξX\X. Let U be an open set in Y , then
f−1(U) is open in X. Find an open set V in ξX such that V ∩ X = f−1(U).
Let W = V ∩ X = f−1(U) and W1 = V ∩ K. Define a topology on ζY by the
collection of all sets of the form f(W) ∪ W1, where W = V ∩ X, W1 = V ∩ K
and V = f−1(U) is an open subset of ξX. We prove that ζY with this topology
is a connectification of Y .

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 4



Hausdorff connectifications

Let V be any open set in ζY . Then there exist an open set U in ξX such
that V = f(U ∩ X) ∪(U\X). We may choose U = f−1(V ) ∩ Y . Since X is
dense in ξX, U ∩ X 6= φ. Thus f(U ∩ X) ∩ Y 6= φ and hence V ∩ Y 6= φ. This
implies that Y is dense in ζY .

If ζY is not connected, there exists a nonempty clopen subset U of ζY such
that U ⊂ ζY and U 6= ζY . Then, there exists an open set V (= f−1(U ∩ Y ))
such that f(V ) = U ∩ Y . Then V ∪ H is an open set in ξX, where H =
U\Y . Similarly, since Y \U is an open set, there exists an open set V1(=
f−1((Y \U) ∩ Y )) such that f(V1) = Y \U. Then V1 ∪ H1 is an open set in ξX,
where H1 = (Y \U)\Y . Trivially, ξX\(V ∪ H) = V1 ∪ H1. Also V ∪ H and
V1 ∪ H1 are both open sets. This proves that ζY is connected.

It remains only to prove that ζY is a Hausdorff space. Let x, y ∈ Y and
x 6= y. If x is in K, let x1 = x and if y is in K, let y1 = y. Otherwise
choose x1 ∈ f

−1(x) and y1 ∈ f
−1(y). Then there are disjoint open sets U

and V in ξX such that x1 ∈ U, y1 ∈ V and int(cl(U)) ∩ int(cl(V )) = φ.
Let U1 = int(cl(U)) ∩ X, U2 = int(cl(U))\X, V1 = int(cl(V )) ∩ X and V2 =
int(cl(V ))\X. Since f seperates every pair of disjoint regular open subsets in
X, we have f(U1) ∩ f(V1) = φ. Then f(U1) ∪ U2 and f(V1) ∪ V2 are disjoint
open subsets of ζY containing x and y, respectively. �

Remark 3.6. In the above theorem 3.5, the condition “f seperates every pair of
disjoint regular open subsets of X” used only for Hausdorffness of a connectifi-
cation. Without that assumption, we can get a non Hausdorff connectification
of a space Y .

Definition 3.7 ([8]). A mapping f : X → Y is a said to be compact mapping
if f−1(x) is a compact subset of X, for everey x ∈ X.

Theorem 3.8. Let f : X → Y be a continuous and compact mapping from X
onto Y . If ξX is any connectification of X, then there is a connectification ζY
for Y

Proof. Proof for “ζX is a connected extension of X” follows from theorem 3.5.
Hausdorffness of ζX is to be proved. Let x, y ∈ Y and x 6= y. K = ξX\X. If
x is in K, let C1 = {x} and if y is in K, let C2 = {y}. Since f is a compact
maping, f−1(x) = C1 and f

−1(y) = C2 are compact subsets of X. Then
there are disjoint open sets U and V in ξX such that C1 ⊆ U and C2 ⊆ V .
Then f(U ∩ X) ∪(U\X) and f(V ∩ X) ∪(V \X) are disjoint open subsets of ζX
containing x and y respectively. �

Definition 3.9. A mapping f : X → Y is said to be a pointly connected
mapping if f−1(x) is a connected subset of X, for every x ∈ Y .

The following examples show that the restriction of a pointly connected
mapping need not be a pointly connected mapping and the composition of
pointly connected mappings need not be a pointly connected mapping.

Example 3.10. Let f : (R+ − Z+) → Z+ ∪{0} be a mapping defined by
f(x) = [x], largest integer less than or equal to x. Then f is a pointly connected

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 5



S. Ramkumar

mapping. Let A = R+ − (Z+ ∪{2n+1
2

: n = 0, 1, 2, · · · · · · }). Then f|−1
A

(m) =

(m, 2m+1
2

) ∪(2m+1
2

, m + 1), which is not a connected subset of A.

Example 3.11. Let f : (R − Z) → Z be a mapping defined by f(x) = [x], for
every x ∈ (R−Z)and g : R−(Z ∪{2n+1

2
: n = 0, 1, 2, · · · · · · }) → R−Z be a map-

ping defined by f(x) = x, for every x ∈ R − (Z ∪{2n+1
2

: n = 0, 1, 2, · · · · · · }).
Here, f and g are pointly connected mappings. But f ◦ g is not a pointly con-
nected mapping, because (f ◦g)−1(m) = (m, 2m+1

2
) ∪(2m+1

2
, m+1), for m > 0,

which is not a connected subset of R − (Z ∪{2n+1
2

: n = 0, 1, 2, · · · · · · }).

Definition 3.12. A mapping f : X → Y is said to be a connected mapping if
f−1(A) is a connected subset of X, whenever A is a connected subset of Y .

Remark 3.13. (1) Restriction of a connected mapping need not be a con-
nected mapping. This can be verified from example 3.10

(2) Composition of connected mappings is a connected mapping.
(3) Every connected mapping is a pointly connected mapping. The con-

verse need not be true.
(4) Let f : X → Y and g : Y → Z be two pointly connected mappings.

Then g◦f is a pointly connected mapping if g is an one to one mapping
or f is a connected mapping.

(5) Let f : X → Y be an one to one mapping from X onto Y . If f is an
open mapping, then f is a connected mapping.

Definition 3.14 ([6]). Let f : X → Y be a mapping from X to Y and U ⊆ X.
Then f is said to be saturated on U, if there is a subset V of Y such that
U = f−1(V )

Theorem 3.15. Let f : X → Y be a mapping from X to Y . If f is an open
mapping on every saturated subset of X and f seperates open subsets of X,
then f is a connected mapping.

Proof. Let U be an connected subset of Y . If f−1(U) is not connected, we can
find a clopen subset V of X such that V ∪X\V = f−1(U). Since f is an open
mapping on every saturated subset of X, f(V ) and f(X\V ) are open subsets
of Y . Also f(V )∪f(X\V ) = U, because f seperates open sets in X, which is a
seperation on U. Thus f−1(U) is connected �

The converse of the above remark 3.13 need not be true. The following
example shows it.

Example 3.16. Let X be the subspace [0, 1] ∪(2, 3] of R and let Y be the
subspace [0, 2] of R. Define a map f : X → Y by

f(x) =

{

x if x ∈ [0, 1]

x − 1 if x ∈ (2, 3]
. Then f is a continuous onto mapping. Also

f is a connected mapping and seperates open subsets of X, but not an open
mapping.

Theorem 3.17. Let f : X → Y be a continuous and connected mapping from
X onto Y . Then X is n-connected if and only if Y is n-connected.

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Hausdorff connectifications

Proof. If f is a continuous and connected mapping, then f preserves connected
subset of X onto a connected subset of Y . �

4. Remainders of Connectifications

Definition 4.1 ([8]). Let ξ1X, ξ2X be two connectifications of a space X.
Then, we write ξ1X ≥ ξ2X, if there is a continuous function f : ξ1X → ξ2X
such that f(x) = x, for all x ∈ X.

Theorem 4.2. Let ξX be any connectification of a space X and {Ki : i =
1, 2, · · ·n} be a collection of mutually disjoint nonempty compact subsets of
ξX\X. Choose n distinct points {pi : i = 1, 2, · · ·n} not in ξX and define

a mapping h : ξX → ζX by h(p) =

{

p if p ∈ ξX\(
n
∪
i=1

Ki)

pi if p ∈ Ki
, where ζX =

(ξX\
n
∪
i=1

Ki) ∪{p1, p2 · · · · · · pn}. Let γX have the quotient topology induced by

h. Then ζX is a connectification for X such that ξX ≥ ζX.

Proof. Proof of this theorem is similar to the proof of Lemma: 2 in [5] �

Theorem 4.3. Let ξX be any connectification of a regular space X. Let {Ki :
i ∈ I} be a collection of mutually disjoint nonempty compact subsets of ξX\X
and they are locally finite in ξX. Let {pi : i ∈ I} be such that pi /∈ X, for every
i ∈ I. Then there is a connectification ζX for X such that ξX ≥ ζX.

Proof. Let ζX = (ξX\ ∪
i∈I

Ki)∪{pi : i ∈ I} and Y = (ξX\ ∪
i∈I

Ki) where pi are

distinct, and pi /∈ ξX. Define a map h : ξX → ζX by h(x) =

{

x if x ∈ Y

pi if x ∈ Ki
.

Let ζX have the quotient topology induced by the map h. Since ξX is con-
nected, ζX is connected. Let U be an open set in ζX. Then h−1(U) is an open
set in ξX which intersects X so that h(h−1(U)) = U intersects h(X) = X.
Hence X is dense in ζX.

Consider two distinct elements y1, y2 in ζX. Let A = h
−1(y1) and B =

h−1(y2). For any x ∈ A, there is an open set Ux of x in ξX such that Ux
intersects only finite number of Ki’s and Ux ∩ B = φ. Define an open set

Wx = Ux\(
n
∪
i=1

Ki) in ξX containing x. Find an open set Vx of x such that

cl(Vx) ⊆ Wx Then {Vx : x ∈ A} is an open cover of A. Since f
−1(x) is compact,

for every x ∈ X. This open cover has a finite subcover {Vx1, Vx2, · · · Vxm}, say.

Let U =
m
∪
i=1

Vxi. Then U is an open set such that A ⊆ U and clU ∩ B = φ.

Similarly, we can find another open set V such that B ⊆ V and U ∩ V = φ.
Then f(U) and f(V ) are disjoint open sets in ζX such that f(A) = y1 ∈ f(U)
and f(B) = y2 ∈ f(V ). This proves the Hausdorffness of ζX. �

Remark 4.4. In the above theorem 4.3, regularity of X is used only for Haus-
dorffness of ζX.

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 7



S. Ramkumar

Definition 4.5 ([8]). A space X is said to be a P-space if every Gδ set is an
open set in X.

Theorem 4.6. Let X be a regular P-space having no isolated points and ξX
be any connectification of X. Let {Ki : i ∈ I} be a collection of mutually
disjoint nonempty compact subsets of ξX\X and they are locally countable in
ξX. Let {pi : i ∈ I} be such that pi /∈ X, for every i ∈ I. Then there is a
connectification ζX for X such that ξX ≥ ζX.

Proof. Proof for connected extension is same as theorem: 4.3. Hausdorffness
of connectification has some difficulties. Consider two distinct elements y1, y2
in ζX. Let A = f−1(y1) and B = f

−1(y2). For any x ∈ A, find an open set
Ux of x such that Ux ∩ Ki 6= φ, for every i ∈ N1 ⊆ N, the set of all natural

numbers. Define an open set Wx = Ux\(
∞

∪
i=1

Ki) in ξX containing x. Find

an open set Vx of x such thatclVx ⊆ Wx. Since f
−1(x) is compact, for every

x ∈ X, {Vx : x ∈ A} has a finite subcollection {Vx1, Vx2, · · · Vxm}, such that

A ⊆ U =
m
∪
i=1

Vxi. Similarly, we can find another open set V such that B ⊆ V

and U ∩ V = φ. Then f(U) and f(V ) are disjoint open sets in ζX such that
f(A) = y1 ∈ f(U) and f(B) = y2 ∈ f(V ). This proves the Hausdorffness of
ζX. �

5. Lattices of Connectifications

Theorem 5.1. If Y is a dense subspace of a space X, then the lattice C(X) of
all connectifications of X can be embedded into the lattice C(Y ) of all connec-
tifications of Y by an order preserving map which also preserves join.

Proof. Since Y is dense in X, every connectification of X is also a connectifi-
cation of Y . �

Theorem 5.2. Let X and Y be two spaces having no isolated points. Let
f : X → Y be a continuous and connected mapping from X onto Y such that
f seperates every pair of disjoint regular open subsets of X. Then the lattice
C(X) is isomorphic to C(Y ).

Proof. If ξX is any connectification of X, then by theorem 3.5, we can find
a connectification ζY for Y with remainder ξX\X. Similarly, if ζY is any
connectification of Y , then there is a connectification ξX for X with remainder
ζY \Y . Thus we have a one to one correspondence from C(X) onto C(Y ).
Let ξ1X and ξ2X be two connectifications of X such that ξ1X ≤ ξ2X, then
there are two connectifications ζ1Y and ζ2Y of Y with remainders ξ1X\X and
ξ2X\X such that ξ1X 7→ ζ1Y and ξ2X 7→ ζ2Y . By our construction in theorem
3.5, we have ζ1Y ≤ ζ2Y . This completes the proof of this theorem. �

Acknowledgements. I am grateful to Prof. C. Ganesa Moorthy, Alagappa

University, Karaikudi for his valuable suggessions to improve this article

c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 8



Hausdorff connectifications

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