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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.

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Hint: A set is open if it is in $\tau$; so, for example $\{a\}$ is open because $\{a\}\in\tau$.

As 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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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.

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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 $\tau$.

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    $\begingroup$ you probably want to change the bold in "The set $A$ is open if $X-A$ is listed...."... edit: do you assume that $A$ is already open? $\endgroup$ – user190080 May 28 '16 at 12:23
  • $\begingroup$ Of course that was supposed to be closed. $\endgroup$ – Daron May 28 '16 at 16:14

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