# Prove that if $x\mapsto -x$ is continuous then $\sigma$ is the discrete topology.

Let $\tau$ be the topology on $\Bbb R$ for which the intervals $[a,b)$ form a base.Let $\sigma$ be a topology on $\Bbb R$ such that $\sigma \supseteq \tau.$

Prove that if $x\mapsto -x$ is continuous then $\sigma$ is the discrete topology.

In order to show that $\sigma$ is the discrete topology we have to show that each singleton $\{x\}$ is open in $(\Bbb R,\sigma)$ .

Please give some hints on how to show that.

• Hint: $(a-1,a]\cap [a,a+1)=\{a\}$. – BigbearZzz May 1 '16 at 4:58

$i(x) = -x$ is its own inverse and for every $x$ we have $(x-1, x] = i^{-1}[[-x, -x+1) ]$, so this is open in any topology like $\sigma$ which makes it continuous.
So is $[x,x+1)$ by virtue of $\tau \subseteq \sigma$, for any $x$.