Prove that the topology T on X is the discrete Topology if and only if {x} $\displaystyle \in$ T for all x $\displaystyle \in$ X.

I know what the definition for discrete topology is but im having a hard time prove this.

Printable View

- Jan 11th 2009, 11:17 AMflamingDiscrete Topology Question
Prove that the topology T on X is the discrete Topology if and only if {x} $\displaystyle \in$ T for all x $\displaystyle \in$ X.

I know what the definition for discrete topology is but im having a hard time prove this. - Jan 11th 2009, 12:08 PMPlato
Recall that a topology is closed under finite intersection and arbitrary union.

If $\displaystyle T$ is a topology on $\displaystyle X$ having the property that $\displaystyle \left( {\forall x \in X} \right)\left[ {\left\{ x \right\} \in T} \right]$

then if $\displaystyle a\;\&\;b$ are two points in $\displaystyle X$ what is $\displaystyle \left\{ a \right\} \cap \left\{ b \right\}$?

What is $\displaystyle \bigcup\limits_{x \in X} {\left\{ x \right\}}$?

If $\displaystyle Y \subseteq X,\quad \bigcup\limits_{x \in Y} {\left\{ x \right\}} = ?$

You said that you know the definition of*discrete topology*.

What does the above show about $\displaystyle T$?