Vtyjkyuk

Let us denote $ V$ to be the set of all real valued functions $ f: mathbbR to mathbbR $. Consider the two Topologies $ tau_1 , tau_2 $ on $ V$ as follows:

$tau_1=$ induced by uniform convergence on compact sets

$tau_2=$induced by local neighborhoods $$ N(f,[a,b], epsilon)=g : $$

Then compare these two topologies $ tau_1, tau_2$.

Are they same?

Answer:

The topology $ tau_1$ induced by uniform convergence while the topology. So it has base $ g(x)-f(x)<epsilon $

which is same as $ tau_2$.

Thus $ tau_1, tau_2$ are same topology .

Am I right ?

Help me doing this

You are right.

(i). If a sequence $(f_n)_n$ converges uniformly to $f$ on every compact subset of $Bbb R$, it will, a fortiori, converge uniformly to $f$ on every compact (i.e. closed bounded) interval.

(ii). If $(f_n)_n$ converges uniformly to $f$ on every compact interval, and if $C$ is any compact subset of $Bbb R,$ then there exist $a,bin Bbb R$ with $[a,b]supset C.$ Now $(f_n)_n$ converges uniformly to $f$ on $[a,b],$ so $(f_n)_n$ converges uniformly to $f$ on any subset of $[a,b].$ In particular, $(f_n)_n$ converges uniformly to $f$ on $C.$