Let $x\in X$.Does there exist a topology $\cal{T}$ on $X$ such that $x$ is a limit point for $\{x\}$?

Question

Let $x\in X$. Does there exist a topology $\cal{T}$ on $X$ such that $x$ is a limit point for $\{x\}$?

My intuition suggests that the answer is no. Here is my attempt to prove the statement:

"Let $(X,\cal{T})$ be a topological space. For every $x\in X$, $x$ is not a limit point for $\{x\}$."

Proof:

Suppose there exists a topology $\cal{T}$ on X such that $x\in X$ is a limit point for $\{x\}$. Let $U$ be any open set containing $x$. Then $(U - \{x\})\cap \{x\} \neq \emptyset$, a contradiction.

Answer 1

Definition of limit of $x$ being a limit point of $\{x\}$: every neighbourhood of $x$ contains a point of $\{x\}$ different from itself. There are no points in $\{x\}$ different from $x$, so it is not possible.

Of course your proof is correct.