open and closed sets in discrete space I am confusing how to determine the set is clopen, neither open or closed, open but not closed and closed but not open. I read an example from "Topology without Tears". 
Let $X=\{a,b,c,d,e,f\}$ and $\tau=\{X,\emptyset,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$. $\tau$ is a topology on $X$. Then 


*

*The set $\{a\}$ is both open and closed.

*The set $\{b,c\}$ neither open nor closed.

*The set $\{c,d\}$ is open but not closed.

*The set $\{a,b,e,f\}$ is closed but not open.


I still cant figure how it will be open,closed,both or neither. Can anyone explain to me? Thank you.
 A: Hint: A set is open if it is in $\tau$; so, for example $\{a\}$ is open because $\{a\}\in\tau$.
A set is closed if its complement is in $\tau$; so, for example, $\{a\}$ is closed because $X\setminus\{a\}=\{b,c,d,e,f\}\in\tau$.
$\tau$ is just a list of the open sets. You can get a list of the closed sets by just writing down the complements of the sets in $\tau$.
A: The set $A$ is open if $A$ is listed as an element of $\tau$. To check just look at the list.
The set $A$ is closed if $X-A$ is listed as an element of $\tau$. To check first calculate $X-A$ then look at the list. 
For example if $A = \{b,c\}$ then $X-A = \{a,d,e,f\}$.
The set $A$ is clopen if both of  $A$ and $X-A$ are listed as elements of $\tau$.
A: Note that every element of $\tau$ is an open set; and every subset $A=X\setminus B$ is closed if $B$ is open in $X$.
For example, In (1), $\{a\}$ is in $\tau$, hence it is open; moreover, $\{a\}=X\setminus \{b,c,d,e,f\}$, and $\{b,c,d,e,f\}$ is open set (since it is in $\tau$), therefore $\{a\}$ is also closed.
Others are similar.
