Does this make S into a one-manifold?

Question

I have the following problem : Let $S =(0,1) \times (0,1) \subset \mathbb{R} ^{2} $ and for each $s,0\leq s \leq 1$ let $V_{s} = {s}\times (0,1)$ and $f_{s} : V_{s} \rightarrow \mathbb{R} ,(s,t) \rightarrow t.$Does this make S into a one-manifold? First I thought of giving a sequence that converges on $V_{s} $ for example is $s=0$ chose the sequence $(n^{-1} )$ To see that it does not converge under the $f_{s} $ function but now I realize that it is like an identity in $\mathbb{R} $ that makes me confused.

Answer 1

If your intent is for the collection $\{(f_s, V_s)\}$ to serve as an atlas on $S$, then you should be more explicit about what topology you are giving $S$. In particular, the sets $V_s$ are not even open in $S$ if it is endowed with the topology it inherits as a subspace of $\mathbb{R}^2$ (or, equivalently, the topology it inherits from the Euclidean metric). The sets $V_s$ are, however, open if you give $S$ the order topology. The problem here is that manifolds are often required to be second countable, but $S$ in the order topology is not second countable.