general topology - Immediate succesors in $[0,1)times mathbb{Z}_+$ and $mathbb{Z}_+times [0,1)$

I am working on one example from Munkres book "Topology"and I would like to clarify one question.

Example: Consider the set $[0,1)$ of real numbers and the set $\mathbb{Z}_+$ of positive integers, both in their usual orders; give $\mathbb{Z}_+\times [0,1)$ the dictionary order. This set has the same order type as the set of nonnegative reals; the function

$$f(n\times t)=n+t-1$$ is the required bijective order-preserving correspondence. On the other hand, the set $[0,1)\times \mathbb{Z}_+$ in the dictionary order has quite a different order type; for example, every element of this ordered set has an immediate successor.

My questions:

1) I've checked that $[0,1)\times \mathbb{Z}_+$ has the same order type as the set of nonnegative reals, right?

2) Any element $(t,n)$ from $[0,1)\times \mathbb{Z}_+$ has immediate successor, namely $(t,n+1)$. Right?

3) But elements in $\mathbb{Z}_+\times [0,1)$ have not immediate successors, right?