# Topology on X

• Sep 24th 2011, 09:14 AM
dwsmith
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{?}\$
• Sep 24th 2011, 09:25 AM
Plato
Re: Topology on X
Quote:

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\}\$
• Sep 24th 2011, 10:19 AM
dwsmith
Re: Topology on X
Quote:

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

But that isn't in topology.
• Sep 24th 2011, 10:37 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 \$\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.