Separating set explained
of
functions with
domain
is called a and is said to (or just) if for any two distinct elements
and
of
there exists a function
such that
[1]
with the topology of
uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by
Marshall H. Stone.
[1] Examples
separates the points of
is a
T1 normal topological space, then
Urysohn's lemma states that the set
of
continuous functions on
with
real (or
complex) values separates points on
is a
locally convex Hausdorff topological vector space over
or
then the Hahn–Banach separation theorem implies that continuous linear functionals on
separate points.
Notes and References
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