Math Is Fun Forum

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Dense and separable

How to prove the space of bounded function on a closed interval  B[a,b]  is non-separable (does not have a countable dense subset) ?
the metric is the sup |g(t)-f(t)| .

I think setting up an uncountable and disjoint collection of subset, and then if a dense set exists, it would contains uncountably many elements.
but I dont know how to do that
or does anyone have a better way?

thanks

benice

Member

Registered: 2010-06-10

Posts: 117

Website

Re: Dense and separable

I think setting up an uncountable and disjoint collection of subset, and then

For each t in [a,b] define the function f_t on [a,b] by
f_t(x) = 1 if x=t and 0 elsewhere.

Consider the set of functions { f_t | t ∈ [a,b] }.