Is my understanding of extreme value theorem correct?

Question

The extreme value theorem says a continuous function achieves its maximum and minimum on a compact domain. Consider $f: (0,1)\rightarrow \mathbb{R}$ with $f(x)=x$. I impose the standard topology on the codoamin. I can choose a trivial topology ($\left\{\varnothing, (0,1)\right\}$) on the domain to make it a compact set. However, now $f$ is not continuous because the preimage of $(0.4,0.6)$, which is an open set in $\mathbb{R}$, is not open in the domain. In sum, whether a function achieves its maximum/minimum should not depend on the topology of the domain/codomain, but the topology does affect whether one can apply the extreme value theorem. Is my understanding correct?

Answer 1

Yes, this is entirely correct. To use a different example illustrating the same point, we can use an example like $f:[0,1]\to\Bbb R$ given by $$ f(x)=\cases{1& if $x=0$\\0& otherwise} $$ Using standard topologies everywhere, we have a compact domain, but a non-continuous function, so you cannot apply the theorem. Yet the function clearly attains both its maximum and its minimum.

You could use a function with the same formula but with all of $\Bbb R$ as its domain instead. This would give you a non-compact domain in addition to a non-continuous function, making it extra clear that the theorem cannot be applied. But not at all changing the fact that the function does attain both its minimum and its maximum.