Bases#
Definitiom: Base for a Topological Space
Let \((X, \tau)\) be a topological space.
A base for \((X,\tau)\) is a collection \(\mathcal{B}\) of open sets such that every open set can be represented as a union of a subcollection of \(\mathcal{B}\).
Warning
This representation is not necessarily unique.
Definition: Subbase
Let \((X, \tau)\) be a topological space.
A subbase for \((X, \tau)\) is a non-empty collection \(\mathcal{S}\) of subsets of \(X\) such that the collection of all intersections of finite subcollections of \(\mathcal{S}\) is a base for \((X, \tau)\).
Definition: Neighborhood Basis
Let \((X, \tau)\) be a topological space and let \(p \in X\).
A collection \(\mathcal{B}_p\) of neighborhoods of \(p\) is a neighborhood basis for \((X, \tau)\) at \(p\) if for each neighborhood \(N(p)\), there exists some \(B \in \mathcal{B}_p\) such that \(B \subseteq N(p)\).
Theorem: Base Criterion
Let \((X, \tau)\) be a topological space.
A collection \(\mathcal{B}\) of open sets is a base for \((X, \tau)\) if and only if for each open set \(U\) and each \(u \in U\), there exists some \(B \in \mathcal{B}\) such that \(B \subseteq U\) and \(u \in B\).
Proof
TODO
Theorem: Topology Generation
Let \((X, \tau)\) be a topological space and let \(\mathcal{B}\) be a base for it.
A subset \(U \subseteq X\) is open if and only if for each \(u \in U\) there exists some \(B \in \mathcal{B}\) such that \(B \subseteq U\) and \(u \in B\).
Proof
TODO
Theorem: Existence of a Topology with a Base
Let \(X\) be a non-empty set and let \(\mathcal{B}\) be a collection of subset of \(X\).
There exists a topology \(\tau_\mathcal{B}\) on \(X\) such that \(\mathcal{B}\) is a base for the topological space \((X, \tau_\mathcal{B})\) if and only if \(X\) is the union of \(\mathcal{B}\) and for each \(B_1, B_2 \in \mathcal{B}\), the intersection \(B_1 \cap B_2\) is a union of a subollection of \(\mathcal{B}\).
Proof
TODO
Countability#
Definition: First-Countability Axiom
A topological space is first-countable if there exists a countable neighborhood base for it at every point.
Definition: Second-Countability Axiom
A topological space is second-countable iff it has a countable base.
Theorem: Second-Countability \(\implies\) First-Countability
If a topological space is second-countable, then it is also first-countable
Proof
TODO