# Topology on X

• Sep 24th 2011, 10:14 AM
dwsmith
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 $\{b\}\mbox{?}$
• Sep 24th 2011, 10:25 AM
Plato
Re: Topology on X
Quote:

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 $\{b\}\mbox{?}$

$\color{blue}X\setminus\{b\}=\{a,c\}$
• Sep 24th 2011, 11:19 AM
dwsmith
Re: Topology on X
Quote:

Originally Posted by Plato
$\color{blue}X\setminus\{b\}=\{a,c\}$

But that isn't in topology.
• Sep 24th 2011, 11:37 AM
Plato
Re: Topology on X
Quote:

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.