Comparison of Topologies - Properties

Properties

Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:

  • τ1 ⊆ τ2
  • the identity map idX : (X, τ2) → (X, τ1) is a continuous map.
  • the identity map idX : (X, τ1) → (X, τ2) is an open map (or, equivalently, a closed map)

Two immediate corollaries of this statement are

  • A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
  • An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.

One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

Read more about this topic:  Comparison Of Topologies

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