Epsilon Neighborhoods And Deleted Neighborhoods

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Definition (Epsilon Neighborhoods And Deleted Neighborhoods)

Given $\varepsilon>0$, an $\varepsilon$-neighborhood of the real number $x$ is the interval $$N_{\varepsilon}(x) = (x-\varepsilon,x+\varepsilon) = \{y\in \mathbb{R}\colon |x-y|<\varepsilon\}.$$ A deleted $\varepsilon$-neighborhood of $x$ is the same interval minus the point $x$, which we'll write as $$N^*_{\varepsilon}(x) = N_{\varepsilon}(x)\setminus\{x\} = (x-\varepsilon,x)\cup(x,x+\varepsilon) = \{y\in \mathbb{R}\colon 0<|x-y|<\varepsilon\}.$$