Catalog of theorems from Topology alongside proofs for each.

Lemma 1. Properties of a Basis (Lemma 1.1.6)
§1.2
Let (X,τ) be a topological space with basis B.If ∅≠U⊆X, then TFAE(a) U∈τ.(b) U is a union of elements in B.That is, U=⋃x∈UBx with Bx∈B.(c) ∀x∈U, ∃B∈B s.t. x∈B⊆U.\text{Let $(X, \tau)$ be a topological space with basis $\mathcal{B}$.} \\ \text{If } \varnothing \neq U \subseteq X \text{, then TFAE} \\[0.5em] \text{(a) } U \in \tau. \\[0.5em] \text{(b) $U$ is a union of elements in $\mathcal{B}$.} \\ \text{That is, } U = \bigcup_{x \in U} B_x \text{ with } B_x \in \mathcal{B}. \\[0.5em] \text{(c) } \forall x \in U, \ \exists B \in \mathcal{B} \text{ s.t. } x \in B \subseteq U.

Spot the cow, a 3D model, morphing into a sphere
Cows are easy to round up