Consider the topological space $(X,\tau)$where the set $X=\{a,b,c,d,e\}$ and $\tau=\{X,\emptyset,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e\}\}$
Determine the limit points of
i) $\{a\}$
ii) $\{b\}$
I altered the exercise in order to expose my doubt in a easier way. The limit point is defined as:$x\in X$ is a limit point of $A\subset X$ if $\forall \mathscr{U},x\in \mathscr{U}\implies A\cap\mathscr{U} $ has a point of $A$ different of $x$.
i) $\{a\}\cap\{a,c,d\}=\{a\}$ which is different from all other points in $\{a,c,d\}$ so $c,d$ would be limit points of {a}.
ii) Regarding ${b}$ the limit points would be $c,d,e$.
Questions:
Is this reasoning right? If not what is the answer?
Where does the definition of limit point comes from? Real line?
Thanks in advance!
$c$ is not a limit point of $\{a\}$ because its open neighbourhood $\{c,d\}$ is disjoint from it. The same holds for $d$.
$e$ is the only limit point of $\{b\}$. It has only one nontrivial neighbourhood and that intersects $b$ (and $b \neq e$).