In 1967, A. Wilansky [
spaces. Also, the notion of αδ-closed sets of a topological space is discussed by R. Devi, V. Kokilavani and P. Basker [2,3]. The concept of slightly continuous functions is introduced and investigated by Erdal Ekici et al. [
in a space 
is αδ-converges to a point 
if 
is eventually in every αδ-open set containing
. Using this concept, we define the αδ-US space, Sequentially-αδ-continuous, Sequentially-Nearly- αδ-continuous, Sequentially-Sub-αδ-continuous and Sequentially-αδO-compact of a topological space
.
and
, respectively. A subset A is said to be regular open (resp. regular closed) if 
(resp.
, the δ-interior [
. The subset A is called δ-open if
, i.e., a set is δ-open if it is the union of regular open sets. The complement of a δ-open set is called δ-closed.
is called δ-closed if
, where
. The family of all δ-open (resp. δ-closed) sets in 
is denoted by 
(resp.
). A subset 
of 
and the complement of a α-open are called α-closed. The intersection of all α-closed sets containing A is called the α-closure of A and is denoted by
, Dually, α-interior of A is defined to be the union of all α-open sets contained in A and is denoted by
.
of a space X is said to be
whenever 
and 
is α-open in 
whenever 
and 
is αg-open in
.
The intersection of all αδ-closed sets of X containing A is called αδ-closure of A and is denoted by
. The union of all αδ-open sets of X contained in A is called αδ-interior of A and is denoted by
.
in a space
, αδ-converges to a point 
if 
is eventually in every αδ-open set containing
.
is said to be αδ-US if every sequence in
, αδ-converges to a point of
.
is said to be
if each pair of distinct points 
and 
in 
there exists an αδ-open set 
in 
such that 
and 
and a αδ-open set 
in 
such that 
and
.
if for each pair of distinct points 
and 
in 
there exists an αδ-open sets 
and 
such that 
and
,
.
.
be an αδ-US-space and 
be two distinct points of
. Consider the sequence
, where 
for any
. Clearly 
αδ-converges to
. Since 
and 
is αδ-US, 
does not αδ-converges to
, i.e., there exists an αδ-open set 
containing 
but not
. Similarly, we obtain an αδ-open set 
containing 
but not
. Thus, 
is
.
-space is αδ-US.
be a 
space and 
a sequence in
. Assume that
αδ-converges to two distinct points 
and
. Then 
is eventually in every 
then 
is eventually in two disjoint αδ-open sets. This is a contradiction. Therefore, 
is αδ-US.
is said to be
.
is an αδ-US space and
is a sequence in the diagonal Δ. It follows that 
is a sequence in
. Since 
is αδ-US, the sequence
αδ-converges to 
which clearly belongs to Δ. Therefore, Δ is a sequentially 
subset of
. Conversely, suppose that the diagonal Δ is a sequentially αδ-closed subset of
. Assume that a sequence 
is αδ-converging to x and
. Then it follows that
αδ-converges to
. By hypothesis, since Δ is sequentially αδ- closed, we have
. Thus
. Therefore, 
is αδ-US.
is αδ-US and a subset M of X is sequentially 
-compact, then M is sequentially αδ-closed.
is any sequence in 
which αδ-converges to a point
. Since M is sequentially αδO-compact, there exists a subsequence 
of
αδ-converges to
. Since 
is αδ-US, we have
. This shows that M is sequentially αδ-closed.
be a family of αδ-US topological spaces with the index set Δ. The product space of 
is denoted by
. Let 
be a sequence in
. Suppose that
αδ-converges to two distinct points x and y in
. Then there exists a 
such that
. Then 
is a sequence in
.
be any αδ-open in 
containing
.
is a αδ-open set of 
is eventually in
. Thus 
is eventually in 
and it αδ-converges to
. Similarly, the sequence
αδ- converges to
. This is a contradiction as 
is a
is
is said to be
if the sequence
αδ-converges to 
whenever a sequence
αδ-converges to
. If 
is sequentially αδ-continuous at each
, then it is said to be sequentially αδ-continuous.
in 
that αδ-converges to
, there exists subsequence 
of 
such that the sequence
αδ-converges to
.
and each sequence 
in αδ-converging tothere exists a subsequence 
of 
and a point
such that the sequence
αδ-converges to
.
of every sequentially αδO-compact set 
of 
is a sequentially αδO-compact subset of
.
and 
be two sequentially αδ-continuous functions. If 
is αδ-US, then the set 
is sequentially αδ-closed.
is αδ-US and 
is any sequence in E that 
-converges to
. Since 
and 
are sequentially αδ-continuous functions, the sequence 
(respectively,
) converges to 
(respectively,
). Since 
for each 
and 
is αδ-US, 
and hence
. This shows that 
is sequentially αδ- closed.
is sequentially sub αδ-US αδ-US continuous if 
is sequentially αδO-compact.
be a sequence in 
that αδ-US converges to
. It follows that 
is a sequence in
. Since 
is sequentially αδO-compactthere exists a subsequence 
of 
that
. Therefore 
is sequentially sub αδ-continuous.
be a sequentially nearly αδ- continuous function and 
be any sequentially αδOcompact subset of
. We will show that 
is a sequentially αδO-compact subset of
. So, assume that 
is any sequence in
. Then for each
, there exists a point 
such that
. Now 
is sequentially αδO-compact, so there exists a subsequence 
of 
that αδ-converges to a point
. Since 
is sequentially nearly αδ-continuous, there exists a subsequence
of 
such that
αδ-converges to
. Therefore, there exists a subsequence 
of 
that αδ-converges to
. This implies that 
is a sequentially αδO-compact set of
.
is a sequentially 
-compact preserving function. Let 
be any point of 
and 
a sequence that αδ-converges to
. We denote the set 
by 
and put
. Since
αδ-converges to
, 
is sequentially αδO-compact. By hypothesis, 
is sequentially αδO-compact subset of
. Now in 
of 
that
. This implies that 
sequentially sub-αδ-continuous.
is sequentially 
-compact preserving if and only if
is sequentially sub-αδ-continuous for each sequentially αδO-compact set 
of
.
is a sequentially αδO-compact preserving function. Then 
is sequentially αδO-compact in 
for each sequentially αδO-compact subset 
of
. Therefore, by Theorem 3.5 
is sequentially sub-αδ-continuous.
be any sequentially αδO-compact set of
. We will show that 
is sequentially αδO-compact subset of
. Let 
be any sequence in
. Then for each
, there exists a point 
such that
. Since 
is a sequence in the sequentially αδO-compact set 
there exists a subsequence 
of 
that αδ-converges to a point in
. By hypothesis
is sequentially sub-αδ-continuous, hence there exists a subsequence 
of 
that αδ-converges to y 
f(M). This implies that f(M) is sequentially αδO-compact in
.
is sequentially sub-αδ-continuous and 
is sequentially αδ-closed in 
for each sequentially αδO-compact set M of
, then f is sequentially αδO-compact preserving.
is sequentially sub-αδ-continuous for each sequentially αδO-compact set 
of 
and by Lemma 3.3. We have already done. So, let 
be any sequence in 
that αδ-converges to a point
. Then, since 
is sequentially sub-αδ-continuous there exists a subsequence 
of 
and a point 
such that
αδ-converges to y.
is a sequence in the sequentially αδclosed set 
of
, we obtain
. This implies that 
is sequentially sub αδ-continuous.
is said to be slightly αδ-continuous if for each 
and for each
, there exists 
such that
, where 
is the family of clopen sets containing f(x) in a space
.
be a directed set 
net 
in 
is said to be αδ-convergent to a point 
if 
is eventually in each 
.
, the following are equivalent:
is slightly αδ-continuous.
for each
.
is αδ-cl-open for each 
CO(Y).
and for each net 
in
.
. Let 
and let 
then
. Since 
is slightly αδ-continuous, there is a 
such that
. Thus
, that is
is a union of αδ-open sets. Hence 
.
. Let
, then
.
.
is αδ-closed.
. Let 
be a net in
αδ-converging to 
and let
. There is thus a
such that
. There is thus a 
such that 
implies 
since
is αδ-convergent to
. Thus
for all
. Thus 
is αδ- convergent to
.
Suppose that 
is not slightly αδ-continuous at a point
, then there exists a
such that 
does not contained in 
for each
. So 
and thus 
for each
, since 
is directed by set inclusion
, there exists a selection function 
from 
into 
for each
. Thus 
is a net in 
αδ-converging to
. Since 
and so
, for each
,
is not eventually in
, which is a contradiction. Hence 
holds.
is slightly αδ-continuous and 
is slightly continuous, then their composition 
is slightly αδ-continuous.
, then
. Since 
is slightly αδ-continuous,
. Thus 
is Slightly αδ-continuous.
:
is slightly αδ-continuous(b) for each 
and for each
there exists αδ-cl-open set 
such that
(c) for each closed set 
of
, 
is αδ- closed(d) 
for each 
and
for each
.
Let 
and
by Theorem 4.3. 
is clopen.
, then 
and
.
is obvious.
since 
is the smallest αδclosed set containing
, hence by
, we have
.
for each
,
. Hence
.
Let
. then
, by
, we have
, since every closed set is αδ-closed, thus
is closed and thus αδ-closed, thus 
and 
is slightly αδ-continuous.
is a slightly αδ-continuous injection and 
is clopen
, then 
is 
.
is clopen
. For any distinct points 
and 
in
, there exist 
such that 
and 
. Since 
is slightly αδ-continuous,
and 
are αδ-open subsets of 
such that 
and 
. This shows that 
is
.
is a slightly αδ-continuous surjection and 
is clopen
, then 
is 
.
and 
in
, there exist disjoint clopen sets U and 
in 
such that 
and
. Since f is slightly αδ-continuous, 
and 
are αδ-open in 
containing 
and 
respectively. Therefore
because
. This shows that 
is
.
and each point
, there exist disjoint open sets 
and 
such that 
and 
.
and 
of
, there exist disjoint open sets 
and 
such that 
and
.
onto a space then 
is clopen regular.
and be
, take
. Since f is slightly αδ-continuous, 
is a αδ-closed set, take
, we have
. Since 
is αδ-regular, there exist disjoint open sets 
and 
such that 
and
. We obtain that 
and 
such that f(U) and f(V) are disjoint open sets. This shows that 
is clopen regular.
is slightly αδ-continuous injective open function from a αδ-normal space 
onto a space
, then 
is cl-open normal.
and 
be disjoint cl-open subsets of 
Since 
is slightly αδ-continuous, 
and
are αδ-closed sets. Take 
and
. We have
. Since 
is αδ- regular, there exist disjoint open sets A and B such that 
and
. We obtain that 
and 
such that 
and 
are disjoint open sets. Thus, Y is clopen normal.