I am watching the below lecture by Dr Tadashi Tokieda on Topology:
https://www.youtube.com/watch?v=J7vojBbvudQ&t=710s
At around the 12:00 minute mark he states that the boundary ($\partial$) of an "ordinary strip"(e.g. a Mobius strip but 'without a twist') is two disconnected circles. This confuses me as I thought the topological boundary of a set was the closure of the set, set minus it's interior.
But surely, the entirety of the strip is not in the interior (as I can't create an open set that does not intersect the space around the strip so no open set is contained in the interior) and therefore the boundary should be the whole set.
Have I misunderstood here? What am I getting wrong?
That's right: every neighborhood of any point on the strip intersects both the strip and its complement in $\mathbb{R}^3$, so as a subset of $\mathbb{R}^3$, the strip is its own boundary.
However, what is meant here is that the strip is seen as a manifold with a boundary. This means that if you encode it as the set of $\bar{x}$ such that $\bar{g}(\bar{x})=0, \bar{f}(\bar{x}) \geq 0,$ the boundary will be the locus $\bar{g}(\bar{x})=0,\bar{f}(\bar{x}) = 0.$ E.g. the boundary of the cylinder $x^2+y^2 = 1, z \geq 0$ is expectedly the circle $x^2+y^2 = 1$ in the plane $z = 0$ Intuitively, this corresponds to the fact that strip's "own" open sets, that is, 2-d discs on it, can lie entirely in the strip when the their center is an inner point, but have to intersect the strip's complement when drawn around the boundary poins.