Editing Base (topology) (section)

==Theorems==

* A topology <math>\tau_2</math> is [[Comparison of topologies|finer]] than a topology <math>\tau_1</math> if and only if for each <math>x\in X</math> and each basic open set <math>B</math> of <math>\tau_1</math> containing <math>x</math>, there is a basic open set of <math>\tau_2</math> containing <math>x</math> and contained in <math>B</math>.
* If <math>\mathcal{B}_1, \ldots, \mathcal{B}_n</math> are bases for the topologies <math>\tau_1, \ldots, \tau_n</math> then the collection of all [[Cartesian product|set products]] <math>B_1 \times \cdots \times B_n</math> with each <math>B_i\in\mathcal{B}_i</math> is a base for the [[product topology]] <math>\tau_1 \times \cdots \times \tau_n.</math> In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
* Let <math>\mathcal{B}</math> be a base for <math>X</math> and let <math>Y</math> be a [[subspace (topology)|subspace]] of <math>X</math>. Then if we intersect each element of <math>\mathcal{B}</math> with <math>Y</math>, the resulting collection of sets is a base for the subspace <math>Y</math>.
* If a function <math>f : X \to Y</math> maps every basic open set of <math>X</math> into an open set of <math>Y</math>, it is an [[open map]]. Similarly, if every preimage of a basic open set of <math>Y</math> is open in <math>X</math>, then <math>f</math> is [[Continuity (topology)|continuous]].
* <math>\mathcal{B}</math> is a base for a topological space <math>X</math> if and only if the subcollection of elements of <math>\mathcal{B}</math> which contain <math>x</math> form a [[local base]] at <math>x</math>, for any point <math>x\in X</math>.