If net $f$ converges to $x$ and $y$ is an accumulation point of $f$ then can it be proved that $f$ converges to $y$?

Question

Let $\mathcal J$ be a directed set, let $X$ be a topological space and let $f:\mathcal J\to X$ be a function.

Then $f$ is a so-called net on $X$.

It can converge to elements of $X$ and can have accumulation points in $X$.

My question:

if $f$ converges to $x$ and $y$ is an accumulation point of $f$ then can it be proved that $f$ converges to $y$?

I am familiar with the following facts:

• $y$ is an accumulation point of $f$ iff there is a subnet of $f$ converging to $y$.

• If $f$ converges to $x$ then all subnets of $f$ converge to $x$.

• The answer to my question is "yes" if $X$ is a Hausdorff space (mainly because in that case there can only be convergence to at most one element).

So it is especially a question about non-Hausdorff spaces.

Answer 1

No. Consider the space $X=\{-1,1\}$ where $\{1\}$ is open. Let your net be $n \in \mathbb{N} \longmapsto (-1)^n \in X$. Then your net converges to $-1$ and not to $1$, but $1$ is an accumulation point.