X + Y where Y is Hausdorff has an at worst removable

THE
AT
WORST
REMOVABLE
AND
DISCONTINUITIES
CLUSTER
OF
FUNCTIONS
SETS
T. R. Hamlett
(received 21 January 1974; revised 25 March 1974)
Abstract.
f :X + Y
A function
x
removable discontinuity at
X-{a;}
Y
where
X
in
iff for every filterbase
8 -*■x , we have that
such that
is Hausdorff has an at worst
/(B)
converges.
8
on
Sufficient
conditions for a function with an at worst removable discontinuity at
x
x
to actually be continuous at
are investigated.
The relationship
between functions which have an at worst removable discontinuity at
and a generalized notion of a cluster set of
1.
X
function from
K
If
Introduction.
into
and
X*
f
at
we have that
p^
X* , then
/
exists.
is a
f
is defined in [3, Definition 3.2]
of points different from
lim/(p^)
is also explored.
are Hausdorff spaces and
p
as having an at worst removable discontinuity at
every sequence
x
x
p
in
X
if for
such that
limp^ = p ,
In this paper we extend this notion by
using filterbases rather than sequences.
We also generalize the concept
of cluster sets and show their relationship to functions which have an
at worst removable discontinuity.
Cl 04)
denote the neighborhood system at
2.
A
will be used to denote the closure of
A function
is a connected subset of
Let
will
x .
f
from a space
X
into a space
said to be connected if for every connected subset
Theorem 2.2.
W(ar)
We will use the following definition and theorems.
Preliminaries.
Definition 2.1.
and
f
C
of
Y
is
X , f(C')
Y .
be a connected mapping of the space
Math. Chronicle 4(1976), 94-100
94
X
into the
T^-space
Y .
If
contained in
Proof.
C
is a connected subset of
X , then
/(C1(C))
is
Cl(,f(C)) •
Observe that it is sufficient for
Y
to be
T
in the proof
given in [3, Theorem 3.3],
Let
Theorem 2.3.
x
be c.ny point of the space
compact subset of the space
set
U
in
Y .
X
and let
K
be a
Suppose that corresponding to each
W(x) , a closed set
F(U )
in
Y
is prescribed in such a
way that
(i)
F(U) c F(V)
(ii)
0(^(1/)
3.
U
in
W(x)
K .
such that
F(U )
does not meet
[5, Lemma, page 436].
At worst removable discontinuity.
A function
Definition 3.1.
Y
does not meet
: £/€N(a:)}
Then there is a set
X
U c v ,
whenever
f
of a space
X
into a Hausdorff space
is said to have an at worst removable discontinuity (abbreviated
a.w.r.d.) at a point
such that
8
x
in
X
if for every filterbase
x , we have that
converges to
lim /(B)
8
X - {x}
on
exists.
It is well known [l, Remark 1, page 213] that the concepts of
convergence based on filterbases and on nets are equivalent.
It
frequently happens, in a given situation, that one of these two methods
for expressing convergence is more convenient than the other.
An
equivalent formulation of Definition 3.1 in terms of nets reads as
follows: A function
/
of a space
an a.w.r.d. at
x
S : D -*■X - {a:}
such that
in
X
X
into a Hausdorff space
D
if for every directed set
S
converges to
Y
has
and net
x , the net
fS:D-+Y.
converges.
Let
Theorem 3.2.
space
Y .
point
y
Then
in
converging to
Y
f
f
be a mapping of the space
has cm a.w.r.d. at
x
in
such that for every filterbase
x , we have that
f(B)
95
X
X
into the Hausdorff
iff there exists a
8
converges to
on
X - {x}
y ,
with
8
f
Proof.
Assume that
and
are filterbases on
A
f (B)
and
Let
Bfl A
Bfl A
x
X . Suppose that
in
X - {ar} which converge to
z
converge to distinct points
{B IM
denote the filterbase
X - {x}
is a filterbase on
f(B nA)
¥
/(A)
has an a.w.r.d. at
f(B HA)
y
and
such that
respectively.
: B t B and A €A} . Clearly,
x , Now
which converges to
z
is a filterbase which accumulates to both
is Hausdorff,
x
B
y , Since
and
cannot converge and we have a contradiction.
Since sufficiency is obvious, the proof is complete.
Example 3.3.
[0,ft]
If we let
f
be the function from the ordinal space
fix') = 1
(see [l]> page 66, Ex, 5) defined by
/(ft) = 0 ,
f
then
is not continuous at
if i / 2
at
ft . Observe that no sequences of points distinct from
to
ft .
Example 3.4.
X
Let
X
topology on
and
ft but does have a.w.r.d.
be the real line and let
T
ft converge
be the smallest
generated by the standard Euclidean topology
S
X
on
together with the topology of countable complements C[4], Ex. 63).
0
Note that a set
is countable.
countable.
f (x) = 0
is open in
f : {X,T)
x < 0 , f(x) = 1
if
p .
0 - U-A
iff
X
Also, observe that
Now let
real number
T
is Hausdorff and
(X,S)
if x > 0 ,
It is easily seen that
f
A
is not first
and
at 0 is not ''removable".
f( 0) = p
for some
vacuously satisfies the
defined at 0 so that it will be continuous.
f
X
and
be the function defined by
requirements for an a.w.r.d. at 0 given in [3],
continuity of
U Z S
where
but cannot be re
In other words the dis
Part 2 of the proof of the
following theorem shows that this problem is eliminated by our defini
tion.
The following theorem is analogous to Theorem 3.6 of [3],
Theorem 3.5.
space
X
in
iff f
X
Proof.
Let
f
be a connected map from the locally connected
into the Hausdorff space
has an a.w.r.d. at
Observe that we may assume
Y . Then
f
is continuous at
x
x .
0 - {x}
is nonempty for every neigh
0
borhood
at
of
is continuous
x . The proof is in two parts.
Part 1.
f
Assume
y f- f{x)
We assume that
U
Let
N(x) - {x}
V
and
respectively.
Let
converges to
neighborhood
0
f(C-{x}) c
X - {x}
y
f(0 - {x}) c JJ ,
W(x)
such that
C
and
Now let
A
x . We may assume that
A - {x}
have that
f(x~) €V
f(A - {x}) c w .
f{A - {x})
If
converges to
If
In either case, we have that
continuous at
X - {x}
/(£)
then
f( A)
is a filterbase on
/(A)
converges to
which converges to
by Part 1.
W
If
in
A ~ {x} = A
x €A ,
= fUA-{x}) U {x»
rily, we conclude that
which converges to
A - {x} = {A - {x} : *4 €A}
= f[A) c W .
f(A)
X
be any filterbase on
x £ A ,
0 . Now we
y = fix') .
f(x) , then there exists an A
neighborhood of
be an open
which contradicts the connectivity
is a filterbase on
/(A-{a;})
/(x)
x . Hence
C
Let
is contained in
; otherwise, we have immediately that
Since
and
which implies that there exists an open
f . Thus we conclude that
Part 2.
y
which converges to
such that
jj
x .
\0 - {x} \ 0€\'(x)},
denote the filterbase
x
of
be the unique
X - {a;} which converge to
be disjoint open neighborhoods of
K'(x) - {x}
/(N(x) - {x})
have
y
and proceed to a contradiction.
is a filterbase on
connected set in
x . Let
has an a.w.r.d. at
limit of all images of filterbases on
of
f
x . Otherwise we have immediately that
A
f(x) .
x } we
is a
such that
and we have
then we have
U {f (x)} c W .
= /U-{x»
f(A) c W . Since
converges to
W
f(x )
was chosen arbitra
and that
f
is
x .
Since necessity is obvious, the proof is complete.
Remark.
It is clear from Part 2 of the above proof that a function
which has a.w.r.d. at a point
re-defined at
define
f(x )
x
x
and is net continuous at
so that it will be continuous at
x .
x
can be
Indeed, re
to be the unique limit of all images of filterbases
97
B
on
X - {s;}
4.
Cluster sets.
such that
B
x .
converges to
The following definition is given in [2, Definition
3] for real valued functions.
If
Definition 4.1.
Y , we say that
y
f
X
is a function from the space
into the space
is an element of the cluster set of
f
at
x ,
C(f;x) , if a: is an element of the derived set of the
denoted
y .
inverse image of every neighborhood of
If
Theorem 4.2.
f
is a function from
X
into
the following are
J ,
equivalent :
(i)
y
(ii)
There exists a filterbase
(iii)
is an element of
B
converges to
y
is an element of
C(f',x) ,
x
and
on
B
f(B )
X - {x}
such that
converges to
y .
fl{Cl(/(£/ - {#})) : J/€W(s:)} .
The proof is straightforward and hence omitted.
Theorem 4.3.
If
the space
into the compact space
Let
Proof.
Since
A
X
I
B
x
is a limit point of
be a filterbase on
V
Now if
/"(B)
J
C(fyx)
subordinate to
is nonempty.
which converges to
f(B )
y
in
x .
X .
Let
which converges to
y .
: U t N(x ) and 4 €A} ,
V
satisfies condition (ii) of Theorem 4.2, and
y €C(f;x) .
Theorem 4.4.
X
Let
f
be a connected mapping of the locally connected
into the T,-space
neighborhood
U
of x
is on element of
Proof.
is a function from
denotes the filterbase
it is easily seen that
space
f
has an accumulation point
{(tf-fo}) fl f 1 ^ )
hence
and
I , then
X - {x}
be a filterbase on
is compact,
X
and
J .
x
If
is connected for every
is a limit point of
X , then
f(x)
C(f;x) .
By Theorem 2.2 we have that
Cl (f(U - { a : } ) )
U - {x}
for every
U
in
/(Cl (U - {x}))
U(x) .
98
is contained in
If we suppose that
f(x)
is
V
not in
C(/;x) , then there exists a
not in
Cl(/(V- {x})) . This implies that
/(Cl(7-{x}))
and consequently
0
exists an
in
W(x)
0[\V = {x} ,
that
x
N(x)
is not in
x
fix')
such that
/(x)
is
is not in
Cl(7-{x}) . Thus there
0 H (7 - {x})
such that
and hence
in
is empty.
We conclude
X
is not a limit point of
in contra
diction to our assumption.
Let
Theorem 4.5.
Hausdorff space
a.w,r,d. at
f
be a mapping of the space
Y , If
x
x iff C(f',x)
f
X
is a limit point of
into the compact
X }
then
f
is a singleton ,
Proof.
Assume that
C(f;x )
is nonempty and by Theorem 3,2 it must be a singleton,
has an a.w.r.d. at
X - {x}
N(x) - {x} ,
V
which converges to
X , the collection
point of
X - {x}
intersect.
does not intersect
conclude that
/(N(x) - {a;})
Thus
/(B)
x
If
converges to
f
y
and
Proof.
which
U
in
Y
does not
such that
f(U - {x}) c V . We
y . Moreover, since
is subordinate to
/
is compact
C(/;x)
M(x)
x . Let
8
/(N(x) - {x}) .
has an a.w.r.d,, at
x .
is a connected map from a locally connected space
into a compact Hausdorff space
f
Y . Since
Y
Y - V . Hence
X - {x} , /(B)
on
converges to
Corollary 4.6.
then
in
By Theorem 2.3, there exists a
converges to
be a
8
x , Since x is a limit
which converges to
is a compact set in
Cl {f{U- {x}))
X
y
be an arbitrary neighborhood of
Y -V
and let
{u - {x} : £/€N(x)} , denoted by
is a filterbase on
and Hausdorff,
x , By Theorem 4,3,
C(f;x ) = {y}
To show sufficiency, assume that
filterbase on
has an
is continuous at
x
Necessity is obvious.
iff
Y
and
C(/;x)
x
is a limit point of
is a singleton .
If we assume that
ton, then Theorem 4.5 implies that
/
C(/;x)
has an a.w.r.d. at
Theorem 3.5, this is sufficient to conclude that
/
is a single
x .
By
is continuous at
x .
Acknowledgement.
The author wishes to thank Paul E. Long for his
patience and guidance.
99
X ,
REFERENCES
1.
J. Dugundji, Topology , Allyn and Bacon, Boston, 1966.
2.
Ulysses Hunter, An abstract formulation of some theorems on
cluster sets , Proc. Amer. Math. Soc. 16(1965), 909-912.
3.
William J. Pervin and Norman E. Levine, Connected mappings of
Hausdorff spaces , Proc. Amer. Math. Soc. 9(1958), 488-496.
4.
Lynn A. Steen and J. Arthur Seebach, Jr, Counterexamples in
Topology, Holt, Rinehart and Winston, Inc., New York, 1970.
5.
J. D. Weston, Some theorems on cluster sets, J. London Math. Soc.
33(1958), 435-441.
University of Arkansas

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