Math Help - Topology on X

1. Topology on X

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

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

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

2. Re: Topology on X

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

3. Re: Topology on X

Originally Posted by Plato
$\color{blue}X\setminus\{b\}=\{a,c\}$
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 $\{b\}$ is not an open set, not in the topology.
There because the complement of $\{b\}$ is not open means $\{b\}$ is not closed.