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2 votes
2 answers
274 views

Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points? For a simple closed curve $\...
S.Zhang's user avatar
  • 23
2 votes
1 answer
200 views

Subset in $[0,1]^k$ with positive density

Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?: For any $A\subseteq\left[0,1\right]^k$ with the measure ...
tom jerry's user avatar
  • 349
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
0 votes
1 answer
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
0 votes
1 answer
80 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
-3 votes
1 answer
230 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
yuta's user avatar
  • 3