# sigma algebra generated by compacts versus sigma algebra generated by open sets

Let $\Omega$ be a locally compact Hausdorff set. Is the sigma algebra generated by compact sets is the same as the sigma algebra generated by open sets?

No. For example, take any non-countable infinite set with the discrete topology. Then the $\sigma$-algebra generated by open sets is all of $\cal P (\Omega)$, but the compact subsets are just the finite subsets, so the $\sigma\text{-algebra}$ generated by compact sets is the collections of countable and countable-complement subsets.
Yes if, say, $\Omega$ is $\sigma$-compact. No in general.
For example let $\Omega = \omega_1$, the set of countable ordinals. If $K$ is compact then $K$ is bounded above (otherwise the intervals $[0,\alpha)$ would be an open conver with no finite subcover). So any compact set is countable. The class of sets $E$ such that either $E$ or $E^c$ is countable is a $\sigma$-algebra, which hence contains the $\sigma$-algebra generated by the compact sets.
But $V=\{\alpha+2:\alpha\in\omega_1\}$ is an uncountable open set such that $V^c$ is also uncountable (the successor of any limit ordinal is in $V^c$).