/sci/ - Science & Math

Let's say that I have open square (square without borders). Distance between points [math]x[/math] and [math]y[/math] is minimum time needet to get from point [math]x[/math] to point [math]y[/math] if you travel with unit speed. This is just Euclidean metric on square.
If we add a portal between points [math]A = (-0.5, 0)[/math] and [math]B = (0.5,0)[/math] that takes some time, let's say 0.5 seconds to get from [math]A[/math] and [math]B[/math] we get some topological manifold.
It is locally Euclidean because every point has neighborhood homeomorphic to [math]\mathbb{R}^2[/math] - for points [math]A[/math] and [math]B[/math] just take ball with radius less than [math]0.5[/math].
Hausdorffnes and countable basis are just as trivial.

By Whitney embedding theorem we know this can be embedded in [math]\mathbb{R}^4[/math].
Can it be embedded in [math]\mathbb{R}^3[/math]?
What is this homeomorphic to?