I was considering the proof about the compactness of the space between rigid unit circles in $\mathbb{R}^3$. Intuitively, I know it is not compact. My way of thinking goes like this and takes two distinct cases: a) the circles lie on the same plane b) the circles are parallel to each other.
Considering the sequence of points that lie on a line passing through the centers of the two circles. As this sequence approaches infinity, it does not have any limit point in the space between the two circles. Therefore, the space is not sequentially compact, and hence not compact.
Another way to see this is to note that the space between the two circles can be continuously deformed into a cylinder without changing its topology. The cylinder is not compact, as it can be continuously deformed into an infinite straight line. Therefore, the space between the two circles is not compact either.
Another approach I thought of, using as its basis the Heine-Borel Theorem goes likes this (a subset of Euclidean space is compact if and only if it is closed and bounded):
First, we can show that the space between two unit circles is not closed. Consider the sequence of points on this space defined by $x_n = (0, 0, n)$. This sequence clearly approaches the $x-y$ plane, which is not contained in the space between the two circles. Therefore, the space is not closed.
Next, we can show that the space is not bounded. Consider the sequence of points on this space defined by $y_n = (n, 0, 0)$. This sequence clearly goes to infinity as n goes to infinity, so the space is not bounded.
Since the space is neither closed nor bounded, it cannot be compact according to the Heine-Borel theorem.
Are these approaches sufficient to conclude my proof? What modifications should I make? Should something else be include?