Let $X$ be a topological space. I know that if $X$ is $T_1$, then every limit point of $Y$ is an $\omega$-accumulation point, and if $X$ is $T_0$, then this does not hold.
So, are there any separation conditions between $T_0$ and $T_1$ such that every limit point is an $\omega$-accumulation point?
My main goal with this is to try to see if there exist weaker conditions than $T_1$ for which Weakly Countable Compactness/Limit Point Compactness is equivalent to Countable Compactness
You need $X$ to be $T_1$.
If $X$ is not $T_1$, there are distinct $x,y\in X$ such that $x$ cannot be separated from $y$, meaning that every open nbhd of $x$ contains $y$. Let $Y=\{y\}$; then $x\in\operatorname{cl}Y$, so $x$ is a limit point of $Y$, but clearly $x$ is not an $\omega$-accumulation point of $Y$, since $Y$ is finite.