Table of Contents

The Closure of a Set in a Topological Space Examples 1
Recall from The Closure of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the closure of $A$ is the smallest closed set containing $A$.
We will now look at some examples of the closure of a set
Example 1
Consider the topological space $(X, \tau)$ where $\tau$ is the discrete topology on $X$. What is the closure of $A \subseteq X$?
Recall that if $\tau$ is the discrete topology then $\tau = \mathcal P (X)$. Hence every subset of $X$ is open and by extension, every subset of $X$ is closed. Hence, for each $A \subseteq X$ the smallest closed set containing $A$ is $A$ itself!
Therefore we have that:
(1)Example 2
Consider the topological space $(X, \tau)$ where $\tau$ is the indiscrete topology on $X$. What is the closure of $A \subseteq X$?
If $\tau$ is the indiscrete topology then $\tau = \{ \emptyset, X \}$. Furthermore, the only closed sets of $X$ with respect to the indiscrete topology are $\emptyset$ and $X$.
If $A \neq \emptyset$ then the smallest closed set containing $A$ is $X$ itself! Hence:
(2)If $A = \emptyset$ then the smallest closed set containing $A$ is $\emptyset$, so:
(3)Example 3
Consider the topological space $(\mathbb{R}, \tau)$ where $\tau = \{ \emptyset \} \cup \{ (n, n) : n \in \mathbb{Z}, n \geq 1 \}$. What is the closure of $A = \{ 0 \}$? What is the closure of $B = (2, 3)$?
Notice that the open sets of $\mathbb{R}$ with respect to the topology $\tau$ are:
(4)Therefore the closed sets of $\mathbb{R}$ with respect to this topology are:
(5)Notice that NONE of these sets except for the whole set $\mathbb{R}$ contain $\{ 0 \}$. Therefore:
(6)Now notice that $(2, 3) \subseteq (\infty, 2] \cup [2, \infty)$. This is the smallest such closed set, and so:
(7)