Limit points of $(0, \ 1]$

Question

I am reading the section in Munkres about limit points, and it says that for the subset $(0, \ 1]$ of the real line, any point of $[0, \ 1]$ acts as a limit point, including, for example, $0.5$. However, if the definition of a limit point is that every neighbourhood containing the limit point intersects the subset at some point other than the limit point itself, couldn't I just define an open set as the interval $(0.5 \ - \ \epsilon, \ 0.5 \ + \ \epsilon)$, for some infinitesimal value $\epsilon$ to contain only the point $0.5$, and therefore not follow the criteria for a limit point? Am I misunderstanding how open intervals can be defined?

Answer 1

Notation. "$(x,y)$" is notation for "$\{\xi\in\mathbb{R}:x<\xi<y\}$."

Terminology. Fix a subset $S$ of $\mathbb{R}$. Fix $x\in\mathbb{R}$. Say $x$ is a limit point of $S$ if for each positive real $\varepsilon>0$, the interval $(x-\varepsilon,x+\varepsilon)$ intersects $S\setminus{\{x\}}$.

Remark. Infinitesimals are not real numbers.