Catalog of Definitions from Topology.

Definition 1. Topology (Def 1.1.1)
§1.1
Let X be a set and let τ be a collection of subsets of X s.t.(1) τ is closed under arbitrary unions:if {Ui:i∈I}⊆τ, then ⋃i∈IUi∈τ.(2) τ is closed under finite intersections:if U1,U2,…,Un∈τ with n∈N, then ⋂k=1nUk∈τ.(3) ∅,X∈τ.Then τ is a topology on X. This can be written as (X,τ)and called a topological space.\text{Let $X$ be a set and let $\tau$ be a collection of subsets of $X$ s.t.} \\[0.5em] \text{(1) $\tau$ is closed under arbitrary unions:} \\ \text{if } \{U_i : i \in I\} \subseteq \tau \text{, then } \bigcup_{i \in I} U_i \in \tau. \\[0.5em] \text{(2) $\tau$ is closed under finite intersections:} \\ \text{if } U_1, U_2, \ldots, U_n \in \tau \text{ with } n \in \mathbb{N} \text{, then } \bigcap_{k=1}^{n} U_k \in \tau. \\[0.5em] \text{(3) } \varnothing, X \in \tau. \\[0.5em] \text{Then $\tau$ is a topology on $X$. This can be written as $(X, \tau)$} \\ \text{and called a \textit{topological space}.}
Definition 2. Open
§1.1
Let (X,τ) be a topological space‾.A subset U⊆X is called open iff U∈τ.\text{Let $(X, \tau)$ be a \underline{\href{/notes/topology/definitions#definition-1}{\text{topological space}}}.} \\ \text{A subset $U \subseteq X$ is called \textit{open} iff $U \in \tau$.}
Definition 3. Closed
§1.1
Let (X,τ) be a topological space‾.A subset C⊆X is called closed iff its complementCc=X∖C∈τ.\text{Let $(X, \tau)$ be a \underline{\href{/notes/topology/definitions#definition-1}{\text{topological space}}}.} \\ \text{A subset $C \subseteq X$ is called \textit{closed} iff its complement} \\ C^c = X \setminus C \in \tau.
Definition 4. Finer / Coarser
§1.1
Let σ,τ be topologies‾ on a set X.We say σ is finer than τ if σ⊇τ.We can also say τ is coarser than σ.Think of it as the space being “chopped up” into finer piecesif σ has more open‾ sets.\text{Let $\sigma, \tau$ be \underline{\href{/notes/topology/definitions#definition-1}{\text{topologies}}} on a set $X$.} \\ \text{We say $\sigma$ is \textit{finer} than $\tau$ if } \sigma \supseteq \tau. \\ \text{We can also say $\tau$ is \textit{coarser} than $\sigma$.} \\[1em] \text{Think of it as the space being “chopped up” into finer pieces} \\ \text{if $\sigma$ has more \underline{\href{/notes/topology/definitions#definition-2}{\text{open}}} sets.}
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.