Trivial Question on Open Set in $\mathbb{R}^2$ with standard euclidean topology

Question

Let $\mathbb{R}^2$ endowed with standard euclidean topology and $U$ be a nonempty open subset of $\mathbb{R}^2$. Fix a point $(x,y)\in U$.

Is it true that for all $(x',y')\in U$ the point $(x',y)$ lies in $U$?

Answer 1

Your intuition about how open sets can behave seems to be way off. Also your modified problem has easy counterexamples:

Your set $U$ is the interior of that obtuse triangle, and you can then always find a point (marked red) whose projection onto the $x$-axis is outside that triangle.

Basically your assumption, that some $(x,0)$ is part of an open set $U$, is incredibly easy to fulfill, and leaves lots of possibilities for $U$. There is no chance at all that any kind of 'refinement' of your question can bring a positive result if it only is based on open sets.

If you are talking about convex sets, then we get nearer, but my example shows that it's still wrong there. So I'd suggest you talk about where this problem comes from, so there can be actual progress made.

Answer 2

Obviously it is not true. Just draw some random shapes on the plane and check. You need to understand that, there's lots of possibilities for the open sets in R². They are not merely about U×V, right?