Topological spaces - construction and purpose - Lec 04 - Frederic Schuller
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as the source of specific proofs, examples and notation they are trying to understand when asking about topology.
Recommended by 0 people across 0 different months, 2020 to 2022.
What it is
Lecture 4 of Frederic Schuller's series on the geometric anatomy of theoretical physics. It introduces topological spaces, covering the definition, the standard topology on R^d, and examples such as the chaotic topology, along with continuity of maps between spaces. It is a university-style blackboard lecture.
What people said
Cites nonuser · 2020-02 · Mathematics Stack Exchange · on “Explanation about a proof regarding the standard topology”
I don't understand the argument at 26:00-27:14 [this video] . Where did he use the definition of the standard $\mathbb{R}^d$ topology? It seems that the empty set is in any family of sets by that proof which is clearly nonsense. A youtube snapshot is helpful - compare the definition of $\mathcal U$ on the left side of the board to the right side where he proves $\emptyset \in \mathcal U$ . [LINK]
Cites FFjet · 2021-01 · Mathematics Stack Exchange · on “Proving $\phi$ is continuous”
In this video [this video] the lecturer gave an example, that is: $\phi : M\to N$ , where $M$ is equipped with arbitrary topology $\mathcal O_M$ , and $N$ is equipped with the chaotic topology $\{\varnothing, N\}$ . Then any $\phi$ is continuous. It's clear that the preimage of $\varnothing$ must be $\varnothing$ and thus a open set in $M$ , but why the preimage of $N$ is an open set in $M$ ?
Cites Dutonic · 2022-08 · Mathematics Stack Exchange · on “What does this symbol mean in set theory? c/=”
Symbol in question [LINK] I've been looking all over for what this means but everything I've seen is just the regular A ⊆ B for A is a subset of B. I'm confused on the difference between these two symbols. Context: Around 4:50 of this introductory lecture on topological spaces [this video] . I just typed this up in photoshop and made a jpg. It should look like ⊆ but with an = instead of - underneath it. Apologies for any confusion.