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ABIM
ABIM
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I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

Edit: @MoisheKohan both solved and refined my initial formulation. Here is the updated version which I initially hoped to express:

Does there exist such a $U_{X_0}$ which is isomorphic to $X_0\times [0,1)$ as manifolds with corners?

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

Edit: @MoisheKohan both solved and refined my initial formulation. Here is the updated version which I initially hoped to express:

Does there exist such a $U_{X_0}$ which is isomorphic to $X_0\times [0,1)$ as manifolds with corners?

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YCor
YCor
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Collar Neighborhood Theoremneighborhood theorem for manifold with Cornerscorners

I was reading this wonderful sequence of posts: https://ncatlab.org/nlab/show/manifold+with+boundary: manifold with boundary and https://ncatlab.org/nlab/show/collar+neighbourhood+theorem: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

Collar Neighborhood Theorem for manifold with Corners

I was reading this wonderful sequence of posts: https://ncatlab.org/nlab/show/manifold+with+boundary and https://ncatlab.org/nlab/show/collar+neighbourhood+theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

Collar neighborhood theorem for manifold with corners

I was reading this wonderful sequence of posts: nlab: manifold with boundary and nlab: collar neighbourhood theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?

Source Link
ABIM
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Collar Neighborhood Theorem for manifold with Corners

I was reading this wonderful sequence of posts: https://ncatlab.org/nlab/show/manifold+with+boundary and https://ncatlab.org/nlab/show/collar+neighbourhood+theorem and I couldn't help but wonder. Is there an extension of the Collar neighborhood Theorem for manifolds with corners; in the sense that if $X$ is a $d$-dimensional manifold with corners then the set $X_0$ of points $x \in X$ with no neighborhood homeomorphic to $\mathbb{R}^d$ has a neighborhood $U_{X_0}$ which is homeomorphic to $X_0\times [0,1)$ such that this homomorphism maps $X_0$ to $X_0\times \{0\}$?