6

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.

tac-eibmoz
  • 161
  • 9
    Take $\mathbb{N}$ with the cofinite topology, and $a_n = n$. Then $(a_n)$ converges to every point. – Daniel Fischer Aug 17 '14 at 15:56
  • I'll have to chew on that for a bit (no experience with the cofinite topology). It seems to me the answer has something to do with the fact that there are distinct points in T1 spaces where the open neighborhoods of the points always have overlap. – tac-eibmoz Aug 17 '14 at 16:24
  • 2
    Right. (And if we're restricting ourselves to limits of sequences, we must avoid that the neighbourhood system is too complicated, first countability makes sequences sufficient for everything.) Perhaps it's a little easier if we take a space with a "doubled point", do you already know the line with two origins? – Daniel Fischer Aug 17 '14 at 16:26
  • Thanks. Hadn't encountered the two origin line before, but that's a good example of a T1 space where sequences that would otherwise converge to 0, instead converge to both origins. – tac-eibmoz Aug 17 '14 at 19:15
  • Please I'd like to ask if the space is 1st countable, will the limit be also unique? – Fareed Abi Farraj Jan 22 '19 at 20:02

1 Answers1

5

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.

amrsa
  • 12,917
tac-eibmoz
  • 161