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Math and science::Topology

Finer and coarser topologies

We say that one topology \( \mathcal{T} \) on \( X \) is finer than another topology \( \mathcal{T}' \) on \( X \) iff [in words...]. \( \mathcal{T}' \) is said to be coarser than \( \mathcal{T} \).

Stronger and weaker are alternative terminology for finer and coarser.

\( \mathcal{T} \) is strictly finer iff [...]

Symbolically,

  • \( \mathcal{T} \) is finer than \( \mathcal{T}' \) ⟺ [...].
  • \( \mathcal{T} \) is strictly finer than \( \mathcal{T}' \) ⟺ [...].