= Diagonal in a product with cofinite and Euclidean topologies
{title2=$\{(t,t):t\in\mathbb R\}\subseteq\mathbb R_{\rm cof}\times\mathbb R\ \cong\ \mathbb R$}
Let $Z$ be the underlying <set> of <real numbers> with the <cofinite topology>. The diagonal $W=\{(t,t):t\in\mathbb R\}$ in $Z\times\mathbb R$ with its <subspace topology> is <homeomorphic> to the usual <real line>. Indeed, $t\mapsto(t,t)$ is a <continuous bijection>: its first coordinate is <continuous> because usual <finite sets> are <closed>, and its second coordinate is the identity. The inverse is the second <coordinate projection> restricted to $W$. Consequently $W$ is not <compact>, whereas $Z$ is <compact> by the fact that <every subset of a cofinite space is compact>. Thus $W$ is not <homeomorphic> to $Z$.
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