Heredity of Lower Separation Axioms on Function Spaces

Njuguna E Muturi

doi:10.4236/apm.2014.43014

Abstract

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 T0, T1, T2 and T3 separation axioms. In this paper, we show that the underlying function space inherits the T0, T1, T2 and T3 separation axioms from the function space, and that these separation axioms are hereditary on function spaces.

Highlights

The set of continuous functions from the space Y to the space Z is denoted by C (Y, Z )
The set open topology τ defined on the set C (Y, Z ) generated by the sets of the form F (U,V ) = { f ∈ C (Y, Z ) : f (U ) ⊂ V }, where the sets U and V ranges over the class C of compact subsets of Y and ΩZ class of open subsets of Z respectively, is called the compact open topology
Endowed with set open topology ζ is written as Cζ ( A, Z ) and is referred to as the underlying function space of the space Cτ (Y, Z )

Summary

Introduction

The set of continuous functions from the space Y to the space Z is denoted by C (Y , Z ). The sets of the form F (U ,V ) forms subbases for the compact open topology τ on C (Y , Z ) (see [1]).

Results

Conclusion