[Math] open and closed sets in discrete space

general-topology

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

  1. The set $\{a\}$ is both open and closed.
  2. The set $\{b,c\}$ neither open nor closed.
  3. The set $\{c,d\}$ is open but not closed.
  4. 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.

Best Answer

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 $\tau$.

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