TITLE: open and closed set in non-metric topology
QUESTION [5 upvotes]: I have just started on learning topology. And I saw this question in Amann and Escher's Analysis I (Exercise 10, page 247): 
Let X:={1,2,3,4,5} and 
$\mathcal{T}:=\{\emptyset, X, \{1\}, \{3, 4\}, \{1, 3, 4\}, \{2,3,4,5\}\}$ 
Determine the closure of {2, 4, 5} 
Since my understanding on the basic concepts are still quite weak, I'd like to ask whether my following reasoning steps are correct or not. Thank you! 

My Reasoning: 
All the sets in T are open sets. 
Also, since $\{2, 3, 4, 5\}$ and $\{1\}$ are complement to each other and they are both elements in the topology. They are clopen sets. 
So the closed sets with regard to this topology are 
$\emptyset, X, \{2,3,4,5\}, \{1\}, \{1, 2, 5\}, \{2, 5\}$ 
Thus the closure of $\{2, 4, 5\}$ is the smallest closed set and at the same time its superset,
that is $\{2, 3, 4, 5\}$

REPLY [2 votes]: You're right about the closure and it's a valid reasoning (easy here because we have a full list of all closed sets). You could also say that $1$ has a neighbourhood $\{1\}$ that misses $\{2,4,5\}$ and so is not in the closure, and every neighbourhood of $3$ also contains $4 \in \{2,4,5\}$ so that $3$ is a limit point of your set, and $1$ is not, and so the closure only adds $3$ to your set. It of course gives the same result..