Recall $(\mathbb{R}, \mathcal{T}_\text{lower limit})$ where lower limit topology $\mathcal{T}_\text{lower limit} = \mathcal{T_\mathcal{B}}$ where
$\mathcal{B} = \{[a,b) \subseteq \mathbb{R}, a < b\}$
Question: Are singleton sets $\{a\}, a \in \mathbb{R}$ open?
Attempt:
$\{a\}^c = \mathbb{R}\backslash\{a\} = (-\infty, a) \cup (a, \infty)
\in \mathcal{T}_{lowerlimit}$So $\{a\}$ is closed
Correct?
Best Answer
You are correct that singletons are closed, but it might be worthwhile to show why exactly $(-\infty, a)$ and $(a, \infty)$ are open for each $a \in \mathbb{R}$. Use sets of the form $[-n, a)$ for $(-\infty, a)$ and something similar for the other. Note that singletons can't be open in this topology. If they were, then what could you say about any arbitrary subset $A \subseteq \mathbb{R}$?