Mixing
Increasing union of regular submanifold
$begingroup$
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 !
differential-geometry
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
$endgroup$
add a comment |
$begingroup$
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 !
differential-geometry
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
$endgroup$
add a comment |
$begingroup$
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 !
differential-geometry
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
$endgroup$
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 !
differential-geometry
differential-geometry
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
asked Jan 7 at 23:29
Noam EluarNoam Eluar
205
205
add a comment |
add a comment |
1 Answer
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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 8 at 0:16
Dan RustDan Rust
22.8k114884
$endgroup$
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 8 at 0:16
Dan RustDan Rust
22.8k114884
$endgroup$
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
add a comment |
$begingroup$
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 8 at 0:16
Dan RustDan Rust
22.8k114884
$endgroup$
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
add a comment |
$begingroup$
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 8 at 0:16
Dan RustDan Rust
22.8k114884
$endgroup$
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 8 at 0:16
Dan RustDan Rust
22.8k114884
answered Jan 8 at 0:16
Dan RustDan Rust
22.8k114884
answered Jan 8 at 0:16
Dan RustDan Rust
22.8k114884
answered Jan 8 at 0:16
Dan RustDan Rust
22.8k114884
22.8k114884
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
add a comment |
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
$begingroup$
Thanks a lot ! And can $cup_n W_n$ be an immersed submanifold ?
$endgroup$
– Noam Eluar
Jan 8 at 0:49
add a comment |
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