Discrete Topology Question

**flaming** · January 11th 2009, 11:17 AM

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

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

**Plato** · January 11th 2009, 12:08 PM

Recall that a topology is closed under finite intersection and arbitrary union.

If $T$ is a topology on $X$ having the property that $\left( {\forall x \in X} \right)\left[ {\left\{ x \right\} \in T} \right]$
then if $a\;\&\;b$ are two points in $X$ what is $\left\{ a \right\} \cap \left\{ b \right\}$ ?

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

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

You said that you know the definition of discrete topology.
What does the above show about $T$ ?

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