1. Introduction
The set of continuous functions from the space 
to the space 
is denoted by
. The set open topology 
defined on the set 
generated by the sets of the form
where the sets 
and 
ranges over the class 
of compact subsets of 
and 
class of open subsets of 
respectively, is called the compact open topology. The sets of the form 
forms subbases for the compact open topology 
on 
(see [1] ). The set open topology 
defined on the set 
generated by the subbases 
where 
and 
is called point open topology (see [2] ).
Let 
for 
family of non-empty open subsets of
. The set 
consist of continuous functions of the form 
where 
is an inclusion mapping (see [3] ).
Let the topological space 
be a 
-space for
, then the function space 
with compact open topology 
inherits the 
-separation axioms for 
(see [4] and [5] ).
Definition 1.1 For
, the sets of the form
as defined in [3] , forms the subbases for point open topology on the set
.
Definition 1.2 The sets of the form
where 
is open in
, 
and
, defines the subbases for the set open topology on the set 
(see [3] ). This topology is referred to as open-open topology (see [6] ). If 
is compact, then 
defines the subbases for the compact open topology on the set
.
The point open topology and the compact open topology are also open-open topologies. The set 
endowed with set open topology 
is written as 
and is referred to as the underlying function space of the space 
(see [3] ).
Definition 1.3 Let 
and 
be open subsets of 
and 
respectively. The set 
forms the subspace of the function space 
with the induced topology 
generated by the subbases 
(see [7] ).
The following lemma and theorem are important for our consideration.
Lemma 1.4 In a regular space, if 
is compact, 
an open subset of a regular space and
, then for some open set
, 
and
.
From the above lemma, the following inference is made. Let 
where 
is a class of compact subsets of 
and
. Then for the space 
with compact open topology
, 
is a compact subset of
. Since 
is a regular space, there exist open sets
, such that 
and
.
This implies that
, in which the assertion 
can be made (see [5] ).
Theorem 1.5 The function 
defined by 
is a homeomorphism (see [7] ).
2. Lower Separation Axioms on the Underlying Function Space 
In this section, we show that the underlying function space 
inherits the 
-separation axioms for 
from the space
. Topologies 
and 
are both compact open.
Theorem 2.1 Let the function space 
be a 
space. The function space 
for 
is a 
space.
Proof. Let 
be distinct maps such that
,
. Then
, 
. For the open set 
containing 
but not 
in
, the open set
in 
contains 
but not
. Therefore the space 
is a 
space. □
Theorem 2.2 Let the function space 
be a 
space. The function space 
for 
is a 
space.
Proof. Let 
be distinct maps such that
,
. Then
,
. For the open sets 
containing 
but not 
and 
containing 
but not 
in
, the open sets
and
in 
are neighborhoods of 
but not 
and 
but not 
respectively. Therefore the space 
is a 
space. □
Theorem 2.3 Let the function space 
be a 
space. The function space 
for 
is a 
space.
Proof. Let 
be distinct maps such that
,
. Then
,
. For the disjoint open sets 
and 
neighborhoods of 
and 
respectively in
, the open sets
and
in 
are disjoint neighborhoods of 
and 
respectively. Therefore the space 
is a 
space. □
Theorem 2.4 Let the function space 
be a regular space for a regular space
. The function space 
for 
is a regular space.
Proof. The space 
is regular for a regular space 
if for the open cover 
of
, there exist open sets 
neighborhoods of 
such that for 
and
, 
for some 
is a neighborhood of 
which does not intersect 
and
. For
, 
implying that
, where
. For 
we have that
, implying that 
and for
,
. Therefore 
is a neighbourhood of 
not intersecting
. 
implies that
. From the assertion 
in Lemma 1.4, we have that
. Therefore 
and 
are two disjoint open sets neighborhoods of 
and 
respectively. Hence the set 
with the induced topology 
is a regular space. □
3. Conclusion
The underlying function space 
inherits the 
-separation axioms for 
from the function space
. From theorem 1.5, the underlying function space 
is homeomorphic to the subspace 
of the function space
. This implies that the subspace 
is a 
-space for
, if the function space 
is a 
-space for
. Therefore the 
-separation axioms for 
are hereditary on function spaces.