I'm starting to learn about manifolds. The definition is: A topological space M is an n-dimensional topological manifold if: 1. M is Hausdorff 2. M is second-countable 3. M is locally Euclidean: each point has a neighborhood homeomorphic to an open subset of ℝⁿ The homeomorphisms φ: U → φ(U) ⊆ ℝⁿ are called coordinate charts. Why do we need the Hausdorff and second-countable conditions? Can someone give an example of a locally Euclidean space that is NOT a manifold because it fails these conditions? And what role do charts play in defining calculus on manifolds?

I am studying discrete mathematics and I need to understand the principle of mathematical induction. I know the basic structure: 1. Base case: Prove P(1) is true 2. Inductive step: Assume P(k) is true, prove P(k+1) is true But I struggle with actual proofs. For example, how would I prove: ∑_(i=1)^(n) i = n(n+1)/2 using induction? And what is the difference between ordinary induction and strong induction? When should I use each? Also, can I prove something like 2ⁿ > n² for n ≥ 5 by induction?

I am studying real analysis and I'm learning about the Baire Category Theorem. It states: In a complete metric space, the intersection of countably many dense open sets is dense. Equivalently: A complete metric space cannot be expressed as a countable union of nowhere dense sets. My questions: 1. What does "nowhere dense" mean? How is it different from "not dense"? 2. What is the intuition behind the Baire Category Theorem? 3. What are some stunning applications of the theorem? 4. How do I prove that ℚ is not a G_δ set using Baire? 5. How does Baire imply that there exist continuous functions that are nowhere differentiable? I want to understand why this theorem is so fundamental.