homeomorphism

1. I Full-cone as topological space

Hello, consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in $E^3$. We can endow it with the topology induced by $E^3$ defining its open sets as the intersections between $E^3$ open sets (euclidean topology) and the full-cone thought itself...
2. I About the definition of a Manifold

Hi, I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in $X$ has an open neighbourhood homeomorphic to the Euclidean space $E^n$. As far as I know the definition of homeomorphism requires to specify a topology for...
3. A Structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. We say that $d_1$ and $d_2$ are equivalent if the identity map from $(X,d_1)$ to $(X,d_2)$ and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
4. I Injective immersion that is not a smooth embedding

Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 $\beta: (-\pi, \pi) \to \mathbb R^2$, with $\beta(t)=(\sin 2t,\sin t)$ As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of $\beta$ is compact in...
5. I Differential structure on a half-cone

Hi, consider an "half-cone" represented in Euclidean space $R^3$ in cartesian coordinates $(x,y,z)$ by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with $R^2$ through, for instance, the projection $p$ of the half-cone on the $R^2$ plane. You can use $p^{-1}$ to get a...
6. Finding homeomorphism between topological spaces

1. Homework Statement show that the two topological spaces are homeomorphic. 2. Homework Equations Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them 3. The Attempt at a Solution I have tried proving that these two spaces are...
7. A Graph homeomorphic to Sphere

Hello, I want to prove that a graph represent a manifold, for this i take the opposites edges of a vertex (edge connected between vertex connected to the current vertex) and this subgraph need to be homeomorphic for example to the 1-sphere if i want a 2 manifold. This criterion ensure that my...
8. Poloidal current in toroidal solenoid

Hi, I'm trying to figure out how the current density for a poloidal current in toroidal solenoid is written. I found you may define a torus by an upper conical ring $(a<r<b,\theta=\theta_1,\phi)$, a lower conical ring $(a<r<b,\theta=\theta_2,\phi)$, an inner spherical ring...