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> Is the set $U(n,\mathbb R)$ of all upper triangular $n\times n$ matrices over $\mathbb R$ a connected set in $M(n,\mathbb R)$ (with its usual topology after identification with $R^{n^2})?$
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I think the answer is **yes** since connectedness is a productive property, $\mathbb R,\{0\}$ are connected and $$U(n,\mathbb R)=\\\mathbb R\times\mathbb R\times...\times\mathbb R\\\times\{0\}\times \mathbb R\times...\times\mathbb R\\...\\\times\{0\}\times\{0\}\times...\times\mathbb R$$

Please tell me whether the attempt is right or wrong!