Heredity of Lower Separation Axioms on Function Spaces

The set of continuous functions from topological space Y to topological space Z endowed with a topology forms the function space. For A subset of Y , the set of continuous functions from the space A to the space Z forms the underlying function space with an induced topology. The function space has properties of topological space dependent on the properties of the space Z , such as the 0 T , 1 T , 2 T and 3 T separation axioms. In this paper, we show that the underlying function space inherits the 0 T , 1 T , 2 T and 3 T separation axioms from the function space, and that these separation axioms are hereditary on function spaces.


Introduction
The set of continuous functions from the space Y to the space Z is denoted by ( ) Let where : i A Y → is an inclusion mapping (see [3]).

Definition 1.1 For A Y
⊂ , the sets of the form as defined in [3], forms the subbases for point open topology on the set ( ) where U is open in A , U ∈C and Z V ∈ Ω , defines the subbases for the set open topology on the set ( ) , C A Z (see [3]).[6]).If U is compact, then ( )  [7]).

This topology is referred to as open-open topology (see
The following lemma and theorem are important for our consideration. This implies that ( ) , in which the assertion ( ) ( ) can be made (see [5]).Theorem 1. 5 The function

Lower Separation Axioms on the Underlying Function Space
be distinct maps such that y Y ∀ ∈ , ( ) ( ) T space.The function space ( ) be distinct maps such that y Y ∀ ∈ , ( ) ( )  ) ( ) is a neighborhood of g which does not intersect ( ) and for x A ∈ , ( ) ( )

Conclusion
The underlying function space N. E. Muturi

,
C Y Z .The set open topo- logy τ defined on the set ( ) , C Y Z generated by the sets of the form U and V ranges over the class C of compact subsets of Y and Z Ω class of open subsets of Z respectively, is called the compact open topology.The sets of the form ( ) , F U V forms subbases for the compact open topology τ on ( ) , C Y Z (see[1]).The set open topology τ defined on the set ( ) is called point open topology (see[2]).

V
defines the subbases for the compact open topology on the set ( ) , C A Z .The point open topology and the compact open topology are also open-open topologies.The set ( ) , C A Z endowed with set open topology ζ is written as ( ) , C A Z ζ and is referred to as the underlying function space of the space be open subsets of Y and Z respectively.The set ( ) topology  generated by the subbases

Lemma 1 . 4
In a regular space, if F is compact, U an open subset of a regular space and F U ⊂ , then for some open set V , F U ⊂ and V U ⊂ .From the above lemma, the following inference is made.Let i K ∈ C where C is a class of compact subsets of Y and i Z U ∈ Ω .Then for the space ( ) , C Y Z τ with compact open topology τ , ( ) i f K is a compact subset of i U .Since Z is a regular space, there exist open sets i Z V ∈ Ω , such that ( )

ζ.Theorem 2 . 1
In this section, we show that the underlying function space Topologies τ and ζ are both compact open.Let the function space a regular space Z if for the open cover

(
with the induced topology ζ is a regular space.□


U V is a i T -space for 0,1, 2,3 i = , if the function space ( ) , C Y Z τ is a i T -space for 0,1, 2,3 i =.Therefore the i T -separation axioms for 0,1, 2,3 i = are hereditary on function spaces.