Take a Hausdorff manifold $M$, with a disjoint boundary $\partial M$ composed of $\partial M_1$ and $\partial M_2$, such that the boundary is compact (I think this doesn't hold for non-compact one, with $\mathbb{R}^2$ with $|y| > x^{-1}$ removed as a counterexample). Is there a collared neighbourhood $\partial M_1 \subset C_1$, $\partial M_2 \subset C_2$ such that $C_1 \cap C_2 = \varnothing$?
The reasonings I've tried usually revolve around taking two collared neighbourhoods, and if there's any overlaps, picking the minimum distance $d$ between any two points of the boundaries and taking $r < d/2$ as the distance to keep from the boundary, but I always run into the trouble that this always involves an infinite number of points, hence I have no guarantee that such a distance won't converge to zero (as can be the case with a non-compact boundary). I tried various ideas of covering the boundaries with open balls, or take an open cover of the boundary with $n-1$ dimensional balls and take some cover of $B^{n-1} \times [O,1)$, and picking finite subcovers to ameliorate the situation, but so far no idea that seems to work correctly, as I'm never quite sure how to show that picking the smallest distance between such sets of points will not just be zero.
Edit : How about this on for size : the manifold is a metric space, hence T5. Pick two disjoint neighbourhoods of the two boundaries, treat them each as individual manifolds, and then give them each a collared neighbourhood. Does that sound alright?
Since $\partial M$ is compact, there is $B_i,\ 1\leq i\leq a+b$, where $B_i$ are copies of $\{ x\in \mathbb{R}^n| |x|<1,\ x_n\geq 0 \}$, s.t. $f_i :B_i\rightarrow M,\ 1\leq i\leq a$ are charts for $\partial M_1$ and $ g_i :B_i\rightarrow M,\ a+1\leq i\leq a+b$ are charts for $\partial M_2$
Define $$ U=\bigcup\ f_i(B_i),\ V=\bigcup\ g_i(B_i)$$ If $U\cap V=\emptyset$, then we are done. Define $$ T:=\min_{x\in g_i^{-1} (U) , \ a+1\leq i\leq a+b}\ \{x_n\} $$
Since $U$ does not meet $\partial M_2$, then $T>0$. So let $V=\bigcup\ g_i(B_i\bigcap \{ x\in \mathbb{R}^n| x_n<\frac{T}{2}\} )$.