### Abstract:

The set of continuous functions from topological space Y to topological space Z when
endowed with a topology forms a function space. This function space inherits properties of
topological spaces from space Z such as the lower separation axioms. Topologies deﬁned on
the function spaces are either splitting or admissible. A topology is splitting if the continuity
of a map deﬁned on the product of topological spaces X and Y to space Z, implies the
continuity of a map defined on a space X to the function space. A topology is admissible if
the continuity of a map deﬁned on a space X to the function space, implies the continuity of a
map deﬁned on the product of spaces X and Y to another space Z. A property of topological
space is hereditary if it is inherited by the subspace and hereditarily if it is found on the
subspace only. Normality is a hereditarily property, the lower separation axioms are
hereditary, while compactness is hereditary with respect to closed subsets. These properties
have not been shown to be hereditary or hereditarily on the subspaces of function spaces.
This research study deﬁnes the underlying set of continuous functions from set A subset of
space Y, to space Z. It shows that the underlying set when endowed with a topology fonns
the underlying function space. Topologies deﬁned on the underlying set are shown to satisfy
the condition of splitting and admissibility and have been named R ACY -splitting topology and
R ACY -admissible topology respectively. The study also shows how properties of topological
spaces, splitting and admissible topologies on the function space relate to those on the
underlying ﬁmction space. To achieve this, properties of continuous functions and set theory
have been used. The results obtained are used to identify hereditary and hereditarily
topological properties on function space and its subspaces.