Increasing union of regular submanifold

Question

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 !

score 0 · Accepted Answer · answered Jan 08 '19 at 00:16

0

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 $S^1 \times S^1$.

answered Jan 08 '19 at 00:16

Dan Rust

30,108

Thanks a lot ! And can $\cup_n W_n$ be an immersed submanifold ? – Noam Eluar Jan 08 '19 at 00:49

Increasing union of regular submanifold

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