Answer to Question #115713 in Real Analysis for Sheela John

Question

Prove that an open interval in R is an open set and a closed intervals is a closed set

Answer 1

1)Let $(a,b)$ be an open interval. Take arbitrary $x\in(a,b)$. We have $B_\delta(x)\subset(a,b)$,where $\delta=\min\{x-a,b-x\}$, so $x$ is an interior point of $(a,b)$.

Since we take arbitrary $x\in(a,b)$, we obtain that every point of $(a,b)$ is an interior point of $(a,b)$, so $(a,b)$ is an open set.

2)Let $[a,b]$ be a closed interval. Take arbitrary $x\in\mathbb R\setminus[a,b]$. We have $B_\delta(x)\subset\mathbb R\setminus[a,b]$, where $\delta=\min\{|x-a|,|b-x|\}$, so $x$ is an interior point of $\mathbb R\setminus[a,b]$.

Since we take arbitrary $x\in\mathbb R\setminus[a,b]$ , we obtain that every point of $\mathbb R\setminus[a,b]$ is an interior point of $\mathbb R\setminus[a,b]$ , so $\mathbb R\setminus[a,b]$ is an open set. By the definition of closed set we have $[a,b]=\mathbb R\setminus(\mathbb R\setminus[a,b])$ is a closed set.