Interesting topologies on $\mathbb{R}$? It is evident in with respect to the usual topology, the sequence $0.9,0.99,0.999,0.9999,\ldots$ converges to $1$.
I am also aware that the same sequence does not converge to $1$ with respect to the lower limit topology.
My question is: Does there exist a topology on $\mathbb{R}$ with respect to which the aforementioned sequence converges to a real number that is NOT $1$?

Take into account that a topology on a set defines its structure, without it $\mathbb R$ is the same as any other set with $\aleph_1$ cardinality. Take any real point $a$ and define the distance in $\mathbb R$ as $d(x,y)=|x-y|$ if $x,y\not\in\{1,a\}$, $d(1,y)=d(y,1)=|a-y|$ and $d(a,y)=d(y,a)=|1-y|$, it is clear that under this topology any sequence that converges to $1$ in the usual topology converges to $a$ in this one.
