topological spaces

1. 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...
2. A The Distance between two points in a hypothetical universe

I have a hypothetical universe where the distance between two points in spacetime is defined as: $$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where $\phi$ has units of $km s^{-2}$. The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski...
3. I Equivalence of Covering Maps and Quotient Maps

I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...
4. I Proving that an action is transitive in the orbits

<Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...
5. Topology: Understanding open sets

1. Homework Statement We define $X=\mathbb{N}^2\cup\{(0,0)\}$ and $\tau$ ( the family of open sets) like this $U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}$ $a)$ Show that $\tau$ satisfies that axioms for...
6. A Topological Quantum Field Theory: Help reading a paper

https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space $Y$ and a spacetime $X$ the author talks about the associated Quantum Hilbert Spaces...
7. Topological space question

1. Homework Statement Hello All, I am experiencing Adventures in Topology. So far, so good, but I have an issue here. In the topological space (Real #s, U), show that 1 is not an element of Cl((2,3]). 2. Homework Equations The closed subsets of our topological space are the converses of...