I was looking at this post.
Is it then true that every point of $\mathbb{R}_{fc}$ the limit point of $[0,1]$? Since $[0.1]$ is infinite? Consider the infinite interval $(7,8)$. This must be clearly open in $R_{fc}$. But if $7.11 \in (7,8)$ is a limit point then the$[0,1] \cap (7,8) = \emptyset$ where $(7,8)$ is A neighborhood of $7.11$ correct? Then $7.11$ cannot be limit point right? what am I missing? Thanks..
Limit points of interval [0,1] in finite complement topology.
general-topology
Best Answer
Just to close the question: $(7,8)$ is not open in $\Bbb R_{fc}$ because its complement is not finite! Recall the definition of that space; the only other open set besides the ones with finite complement is $\emptyset$ (because it must be open always).