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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.