Basic Topology: Closure, Interior, Exterior, and Boundary of Set A in the Usual Topology

I am trying to understand this problem and apply the basic definitions given in the text. Please let me know if I am misunderstanding this.

Let $A = [1,2) \cup (3,4)\cup \{5\}$ be a subset of $(\mathbb{R},U)$ and $U$ is the Usual topology. Find each of the following:

(a) $\operatorname{Cl}(A)$.

For this I have $[1,2] \cup [3,4] \cup {5}$ which is the smallest closed set that will contain $A$ in the Usual Topology.

(b) $\operatorname{Int}(A)$.

For this I have $(1,2) \cup (3,4)$, which is the largest open subset of $A$.

(c) $\operatorname{Ext}(A)$.

For this I have $(-\infty,1) \cup (2,3) \cup (4,5) \cup (5,\infty)$, which is based on the Theorem that if $A$ is a subset of $X$, then $\operatorname{Ext}(A) = \operatorname{Int}(X-A)$.

(d) $\operatorname{Bd}(A)$.

For this I have $\{1,2,3,4,5\}$, which is based on $\operatorname{Bd}(A) = X - \operatorname{Int}(A) - \operatorname{Ext}(A)$.

• This looks fine to me. – Captain Lama Apr 11 '16 at 15:56
• Another option for (c): $\text{Ext}(A)$ equals the complement of $\text{Cl(A)}$ – drhab Apr 11 '16 at 16:37
• @drhab thanks. I will remember that as well. – NUG Apr 11 '16 at 16:59