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Nate River
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Notation: 

We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Question:

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

Notation: 

We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Question:

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

deleted 1 character in body; edited title
Source Link
Nate River
Nate River
  • 6.3k
  • 2
  • 23
  • 100

Existence of a certain foliation onof $\mathbb R^n$

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$, with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

Existence of a certain foliation on $\mathbb R^n$

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$, with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

Existence of a certain foliation of $\mathbb R^n$

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.

Source Link
Nate River
Nate River
  • 6.3k
  • 2
  • 23
  • 100

Existence of a certain foliation on $\mathbb R^n$

Notation: We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$, with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.