Give an example of a locally compact Hausdorff space which is not compact.
Would the following be valid? Could someone give an example using a different topology? $(\mathbb{R}, \tau_{\text{euclidian}})$?
I have proved $(X,\tau_{\text{discrete}})$ is locally compact and Hausdorff.
Lets take $(\mathbb{R}, \tau_{\text{discrete}}$, then this is locally compact and Hausdorff, but not compact.
Let $[0,1] \subseteq \mathbb{R}$ and consider the open cover $\bigcup_{x\in [0,1]} \{x\}$. This cover does not contain a finite subcover, so $\mathbb{R}$ is not compact.