TITLE: Is the set $U(n,\mathbb R)$ of all upper triangular $n\times n$ matrices over $\mathbb R$ a connected set in $M(n,\mathbb R)?$
QUESTION [7 upvotes]: 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 $R^{n^2})?$

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!

REPLY [2 votes]: It is more, They are path connected too!
According to your notation, $A,B\in U(n,\mathbb{R})$, The continous path from $[0,1]$ is $f(t)=tA+(1-t)B$