Editing Topological space (section)

== Comparison of topologies ==
{{main|Comparison of topologies}}

Many topologies can be defined on a set to form a topological space. When every open set of a topology <math>\tau_1</math> is also open for a topology <math>\tau_2,</math> one says that <math>\tau_2</math> is {{em|finer}} than <math>\tau_1,</math> and <math>\tau_1</math> is {{em|coarser}} than <math>\tau_2.</math> A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology.  The terms {{em|larger}} and {{em|smaller}} are sometimes used in place of finer and coarser, respectively.  The terms {{em|stronger}} and {{em|weaker}} are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.

The collection of all topologies on a given fixed set <math>X</math> forms a [[complete lattice]]: if <math>F = \left\{ \tau_{\alpha} : \alpha \in A \right\}</math> is a collection of topologies on <math>X,</math> then the [[Join and meet|meet]] of <math>F</math> is the intersection of <math>F,</math> and the [[Join and meet|join]] of <math>F</math> is the meet of the collection of all topologies on <math>X</math> that contain every member of <math>F.</math>