I need to show that $X = [0,1]\subset \mathbb{R}_l$ in the subspace topology, where $\mathbb{R}_l$ is the lower limit topology, is not compact in the limit point definition, which is:
"a topological space $X$ is said to be limit point compact if every infinite subset of $X$ has a limit point in $X$"
So I need to show an infinite subset of $X$ which has a limit point not in $X$. I know that the open sets in $\mathbb{R}_l$ are of the form $[a,b)$. However, I can't imagine a subset of $X$ such that its limit poits are going to be outside it. It seems impossible.
Best Answer
HINT: Find an infinite subset of $[0,1]$ whose only limit point in the usual topology is $1$.