Here's a question I've asked myself : let $M$ be some manifold, and let $W_0 \subset W_1 \subset W_2 \dots $ be countable increasing sequence of regular submanifolds of $W$. Is $\cup_n W_n$ a regular submanifold of $W$ ? I don't think this is true, but I cannot find a counterexample...
Any help greatly appreciated !
Just take $$(S^1 \times \{1\}) \cup \bigcup_{n \in \mathbb{N}}( S^1 \times \{e^{\frac{\pi}{n}i}\})$$ which are each individually submanifolds of the 2-torus $S^1 \times S^1$.