Catalog of Definitions from Topology.

Definition 5. Basis
§1.2
Let (X,τ) be a topological space‾.B⊆τ is called a basis of τ if∀x∈X and ∀U∈τ with x∈U, ∃B∈B s.t. x∈B⊆U.Note that if B is a basis for (X,τ), then:(B1) ∀x∈X ∃B∈B s.t. x∈B.(B2) ∀B1,B2∈B and ∀x∈B1∩B2,∃B3∈B s.t. x∈B3⊆B1∩B2.(B1) follows from the def of basis with U=Xand (B2) from the def of basis with U=B1∩B2.\text{Let $(X, \tau)$ be a \underline{\href{/notes/topology/definitions#definition-1}{\text{topological space}}}.} \\ \mathcal{B} \subseteq \tau \text{ is called a \textit{basis} of $\tau$ if} \\ \forall x \in X \text{ and } \forall U \in \tau \text{ with } x \in U, \ \exists B \in \mathcal{B} \text{ s.t. } x \in B \subseteq U. \\[1em] \text{Note that if $\mathcal{B}$ is a basis for $(X, \tau)$, then:} \\[0.5em] \text{(B1) } \forall x \in X \ \exists B \in \mathcal{B} \text{ s.t. } x \in B. \\[0.5em] \text{(B2) } \forall B_1, B_2 \in \mathcal{B} \text{ and } \forall x \in B_1 \cap B_2, \\ \exists B_3 \in \mathcal{B} \text{ s.t. } x \in B_3 \subseteq B_1 \cap B_2. \\[0.5em] \text{(B1) follows from the def of basis with } U = X \\ \text{and (B2) from the def of basis with } U = B_1 \cap B_2.