Lecture 2: Topological Manifolds (International Winter School on Gravity and Light 2015)

What people said

Recommends Nicolas Lussier-Clément · 2018-03 · Mathematics Stack Exchange · on “Manifolds and open sets in them with different dimensions”

Just to warn, I'm fairly new to this discipline so you might verify further. From Lecture 1 of Gravity and light [LINK] and books, I understood that a topological space is a set $M$ equip with a topology $\mathscr{O}_M)$ that is a subset of the power set of $M$, $(M,\mathscr{O}_M)|\mathscr{O}_M\subset\mathscr{P}(M)$. The topology $\mathscr{O}_M)$ must satisfy 3 additional axiom: $\varnothing, M \in\mathscr{O}_M$ $U,V\in \mathscr{O}_M \Rightarrow U \cap V \in \mathscr{O}_M$ $U_\alpha\in \mathscr{O}_M \Rightarrow \left( \bigcup\limits_{\alpha \in A} U_\alpha \right) \in \mathscr{O}_M$ Meaning that the topological space does not carry dimensions and in that sense you might be able to define suc

Recommends marc · 2018-05 · Mathematics Stack Exchange · on “d-dimensional topological manifold example”

I am having trouble understanding the concepts of manifolds, so I recently watched these two videos on the topic topology [LINK] manifolds [this video] . During watching I came up with the following example, for which I will briefly denote the required definitions I am trying to grasp first. The definition in question given for a $d$-dimensional topological manifold is: Definition : A topological space $(M, \mathcal{O})$ is called a d-dimensional topological manifold , if $$ \forall p \in M : \exists \mathcal{U} \in \mathcal{O} : \exists x : \mathcal{U} \to x(\mathcal{U}) \subseteq \mathbb{R}^d,$$ such that $x$ is invertible: $x^{-1} : x(\mathcal{U}) \to \mathcal{U}$ $x$ is continuous $x^{-1}$ is

Recommends Antoni Parellada · 2021-05 · Mathematics Stack Exchange · on “Where does the metric enter the picture in the definition of the standard topology?”

Here is the exact moment [this video] where this is discussed. I think the video is excellent, but I am just not presenting my question properly.

Recommends Ben H · 2022-12 · Mathematics Stack Exchange · on “Could you give me advice on self-study for topology, measure theory and functional analysis?”

Check out the lectures by Frederic Schuller. You can watch the first two in his GR lecture series, on Topology [LINK] and on Topological Manifolds [this video] , to see if you like his style, and then try Lectures on Quantum Theory [LINK] , which are highly mathematical, following von Neumann's functional analysis approach (and have a standalone lecture on Measure Theory [LINK] ). Paired with appropriate textbooks for further reference, they're perfect.

Cites marcoqc · 2023-09 · Mathematics Stack Exchange · on “Bifurcations and topological manifolds”

I'm following an online course on Differential Geometry by F. Schuller and in this lesson [this video] , around 23:00, he draws a bifurcation and says that isn't a topological manifold because any open set around the bifurcation can't be mapped continuously and with continuous inverse to $\mathbb{R}$ , $\mathbb{R}^2$ , $\mathbb{R}^3$ , etc. [LINK] I don't understand why. If we take $\mathbb{R}^2$ , I can take every point of the bifurcation and map it to a curve in $\mathbb{R}^2$ (so, a section of $\mathbb{R}^2$ ) that looks like a Y. Similarly, I can take any point of that curve and map it back to the manifold. What is the problem here? I get why it isn't a differentiable manifold, but why not a

Critical marc · 2018-05 · Mathematics Stack Exchange · on “d-dimensional topological manifold example”

@freakish Okay. got it. Unfortunately there was this very misleading part [this video] of the lecture, where I got the incomplete definition from. So basically I assume I have to require as you said the target of the chart to be $x(\mathcal{U}) = W\in \mathcal{O}_\text{standard}$, which makes intuitively sense when working with continuity in real spaces, as it gives the known "neighbourhoods" of elements. Of course this might add a number of points to the target set, but then their preimages are the empty set.