Let $(X, \tau) $ be a topological space. If $(X, \tau) $ is Lindel$\ddot{\text{o}}$f then any uncountable subsets of $X$ has a limit point.
I want to know about the converse.
From here , i came to know about an important class of topological spaces( separable Banach spaces) where every uncountable sets has a limit point.But here it doesn't help us to provide any new information as any separable Banach space is again a Lindel$\ddot{\text{o}}$f.
My question : Suppose any uncountable subsets of $X$ has a limit point . Does this implies $(X, \tau) $ is lindel$\ddot{\text{o}}$f ?
Best Answer
The answer to your question is no. The answer to this post shows that $\mathbb{R} \times \omega_1$ has the property that every uncountable subset has a limit point. However, $\mathbb{R} \times \omega_1$ is not Lindelöf because it continuously maps onto $\omega_1$, continuous surjections preserve being Lindelöf, and $\omega_1$ is not Lindelöf.