Let us say that there is Point $1$. And there are Points $2$, $3$ and $4$. ($X$ = $\{1,2,3,4\}$.)
And then, $\tau$ of topological space $X$ is defined as $\{\emptyset, \{1\}, \{1,2,3,4\}\}$.
The first question is, as $\{1,2,3,4\}$, $1$ would be able to reach $2$, which is in the set. Then, why is $\{1,2\}$ not automatically included in $\tau$?
The second question is, what would $\{1\}$ being an open set mean? $\{1\}$ would mean that it is possible to move somewhat and reach itself, as far as I know, and it does not seem to make sense.
I think I am getting some concepts wrong during the class, so can anyone help me correct misunderstanding?
Edit: OK, I now get some points. So, say there is a sequence $A$. The topological space is $X$, and $\tau = \{\emptyset,\{1\},\{1,2\},\{1,2,3,4\}\}$. If $A$ converges to $2$, it would converge to $1$ and $3$ and $4$. Then, why is $\{1,2\}$ needed after all? Does this mean that there is a sequence that converges only to $2$, and not $3$ and $4$?