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...
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...
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...
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...
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...
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...
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...
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...