1. ## Topology on X

Consider the set $\displaystyle X=\{a,b,c\}$

The collection of subsets are $\displaystyle \O,X,\{a,b\},\{b,c\},\{b\}$

The book says the one point set $\displaystyle \{b\}$ is not closed because its complement is not open. What is the complement of $\displaystyle \{b\}\mbox{?}$

2. ## Re: Topology on X

Originally Posted by dwsmith
Consider the set $\displaystyle X=\{a,b,c\}$
The collection of subsets are $\displaystyle \O,X,\{a,b\},\{b,c\},\{b\}$
The book says the one point set $\displaystyle \{b\}$ is not closed because its complement is not open. What is the complement of $\displaystyle \{b\}\mbox{?}$
$\displaystyle \color{blue}X\setminus\{b\}=\{a,c\}$

3. ## Re: Topology on X

Originally Posted by Plato
$\displaystyle \color{blue}X\setminus\{b\}=\{a,c\}$
But that isn't in topology.

4. ## Re: Topology on X

Originally Posted by dwsmith
But that isn't in topology.
That is exactly the point.
The complement of $\displaystyle \{b\}$ is not an open set, not in the topology.
There because the complement of $\displaystyle \{b\}$ is not open means $\displaystyle \{b\}$ is not closed.