Complete Lattice: Topological Space

Complete Lattice: Topological Space
Comparison of topologies
Main article: Comparison of topologies
A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also in a topology τ2 and τ1 is a subset of τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. 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 larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and 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 X forms a complete lattice: if F = {τα | α ∈ A} is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X that contain every member of F.

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