I have zero clues on how to solve this:
"Let be R with the discrete topology d, and R with the eucledian topology e, and X=R x R with the product topology t=d x e. X with the topology product t is a Hausdorff, locally compacted space. Determine the compact sets of X".
I've tried to use Tichonov, but it doesn't cover all the compact sets of X, just the ones that are a product of compacte sets. I tried to find a norm that could induce the topology product on X, so i could use othe theorems about finite dimensions spaces, but I struggle with the discrete topolgy that can't be induced by any norm. Any help?
Let $K$ be a compact subset of $\Bbb R^2$m endowed with that topology. If $(x,y)\in\Bbb R^2$, let $\pi_1(x,y)=x$. Since $\pi_1$ is continuous, $\pi_1(K)$ is compact, with respect to the discrete topology. Therefore, $\pi_1(K)$ is finite. So, $K=\{x_1\}\times K_1\cup\{x_2\}\times K_2\cup\ldots\cup\{x_n\}\times K_n$, for some subset $\{x_1,x_2,\ldots,x_n\}$ of $\Bbb R$ and some subsets $K_1,\ldots,K_n$ of $\Bbb R$. But each $\{x_j\}\times\Bbb R$ is a closed subset of $\Bbb R$, therefore $\{x_j\}\times K_j$ is compact, since it is equal to $(\{x_j\}\times\Bbb R)\cap K$. Therefore, each $K_j$ is compact.
So, the answer is: $K$ is compact, if and only if it can be written as$$\{x_1\}\times K_1\cup\{x_2\}\times K_2\cup\ldots\cup\{x_n\}\times K_n,$$for some compact subsets $K_1,\ldots,K_n$ of $\Bbb R$.