Advances in Pure Mathematics
Vol.4 No.3(2014), Article ID:44351,4 pages DOI:10.4236/apm.2014.43014
Heredity of Lower Separation Axioms on Function Spaces
Njuguna E. Muturi
Department of Mathematics, Egerton University, Egerton, Kenya
Email: edward.njuguna@gmail.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 23 January 2014; revised 23 February 2014; accepted 28 February 2014
ABSTRACT
The set of continuous functions from topological space to topological space
endowed with a topology forms the function space. For
subset of
, the set of continuous functions from the space
to the space
forms the underlying function space with an induced topology. The function space has properties of topological space dependent on the properties of the space
, such as the
,
,
and
separation axioms. In this paper, we show that the underlying function space inherits the
,
,
and
separation axioms from the function space, and that these separation axioms are hereditary on function spaces.
Keywords:Function Space; Underlying Function Space; Hereditary Properties
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.
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