Open set of the torus containing a point of the equivalence class?

Consider the torus defined by the quotient topology $S^1times[0,1];/sim$ where $sim$ defines the equivalence relation with $(x,0)sim(x,1)$ for any $xin S^1$. I need to characterise the open subsets of the torus containing the point $[(x,1)]$, which is the equivalence class ${(x,0),(x,1)| xin S^1}$. Also, the question gives a hint that we can make use of the Sierpinski space.

The steps I have done so far is to consider the equiv. classes as the product of a point on the 1-disk and the set $X = {0,1}$ i.e. ${x}times X$ and I think I need to somehow apply the Sierpinski topology here and link the open sets of the torus and space… (which is the only thing I could think of right now)

Would anyone give me some hint on this? Thanks!