Hi everyone I was reading Dudley's book and I'm having problems with the following.
If $X$ is uncountable, show that there is a net of indicator functions of finite set converging pointwise to the constant function $1$, and also the net cannot be replaced by a sequence.
My idea was to use the set $\Lambda=\{A\in 2^{X}: A \text{ finite }\}$ partially ordered by set inclusion as the directed set, i.e, $A\le B \iff A\subset B$. So define the net $\{1_A\}_{A\in \Lambda}$ But from here I'm not completely sure of how to show that this converges to $1$, because Dudley at this point never defines the pointwise converges. I think that is sufficient to show that for any nbhd of $x$ there is some $A\in \Lambda$, s.t. for $B\ge A$ we have $1_B\to 1$ but this is easy since we can considere $A= \{x\}$ and all the finite subset and this implies that $1_B \to 1$
Any idea? Am I totally out of track?
Thanks