Consider the following set in the dictionary order. Which are linear continua?
(1) $\mathbb{Z_+}\times [0,1)$
(2) $[0,1)\times \mathbb{Z_+}$
my answer :$1) $is true since, for $x\neq y$, ($n\times x,m\times y$) always contains the points $n\times (\dfrac{x+y}{2})$ and $m\times (\dfrac{x+y}{2})$.
for answer $2)$:is true since, for $x\neq y$, ($x\times n,y\times m$) always contains the points $(\dfrac{x+y}{2})\times n $ and $(\dfrac{x+y}{2})\times m$
Is my answer is correct ?
pliz verify its
No, in the second one, there are no points between $(a,n)$ and $(a,n+1)$ for any $a\in[0,1)$ and $n\in \mathbb Z_+.$