TITLE: Unique Limits in T1 Spaces
QUESTION [5 upvotes]: It's intuitive to me that limits in T2 (Hausdorff) spaces are unique: $x_n \rightarrow l$ if you can find an $N$ such that for $n > N$, $x_n \in O$ where $O$ is any open neighborhood of $l$ and since distinct points in T2 spaces always have disjoint open neighborhoods, it's clear that the sequence will "zero in" on a unique point since you can pick an open neighborhood of $l$ that excludes any other point in question.  
I have a vague understanding of why this intuition doesn't work in T1 spaces, but would someone be able to explicitly show why sequences in T1 spaces do not necessarily have unique limits? Simple examples would be helpful. Thanks.

REPLY [3 votes]: To answer my own question, if $x$ and $y$ are distinct points in a T1 space, there are open neighborhoods $O_x$ and $O_y$ such that $x \notin O_y$, $y \not\in O_x$, but where it's possible that $O_x \cap O_y \neq \emptyset$ for all $O_x$, $O_y$. Thus a sequence $a_n$ can converge to both $x$ and $y$: if for $n > N_x$, $a_n \in O_x$ and for $n > N_y$, $a_n \in O_y$, then for $n > \max\{N_x, N_y\}$, $a_n$ is in both $O_x$ and $O_y$, which is allowed in T1 spaces. In T2 spaces, $O_x$ and $O_y$ can be chosen to be disjoint, which is why limits are unique in T2 spaces.