Example of Set X which is not well ordered such that $X\times [0,1)$ with dictionary ordered is not Linear Continuum
I had proved following theorem
Set X which is not well ordered then $X\times [0,1)$ with dictionary ordered is Linear Continuum.
Linear Continuum have the property of LUB and for any 2 point of its there is point between them.
I wanted to know is there converse true or not?
I found all example till that are well ordered
But I was suspicous about converse .
Please give me hint to construct a counterexample
Any Help will be appreciated
If $X \times_{\textrm{lex}} [0,1)$ is a continuum, $X$ must have the property that a non-maximal element has a successor $x^+$: What else can the supremum of $\{x\} \times [0,1)$ else be but $(x^+,0)$, where $(x,x^+)\subseteq X$ is empty? This property is implied by being a well-order, but it might be almost equivalent to what you want. $X=\mathbb{Z}$ is an example: note that $\mathbb{Z} \times_{\textrm{lex}} [0,1)$ is homeomorphic to $\mathbb{R}$, a continuum (via $x \to (\lfloor x\rfloor, x - \lfloor x\rfloor )$, e.g., while $\mathbb{Z}$ is not well-ordered.