nLab finer topology




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory




(finer/coarser topologies)

Let XX be a set, and let τ 1,τ 2P(X)\tau_1, \tau_2 \subset P(X) be two topologies on XX, hence two choices of open subsets for XX, making it a topological space. If

τ 1τ 2 \tau_1 \subset \tau_2

hence if every open subset of XX with respect to τ 1\tau_1 is also regarded as open by τ 2\tau_2, then one says that

  • the topology τ 2\tau_2 is finer than the topology τ 1\tau_1

  • the topology τ 1\tau_1 is coarser than the topology τ 2\tau_2.



(discrete and co-discrete topology)

Let XX be any set. Then there are always the following two extreme possibilities of equipping XX with a topology τP(X)\tau \subset P(X), and hence making it a topological space:

  1. τP(X)\tau \coloneqq P(X) the set of all subsets;

    this is called the discrete topology on XX, it is the finest topology (def. ) on XX,

    we write Disc(X)Disc(X) for the resulting topological space;

  2. τ{,X}\tau \coloneqq \{ \emptyset, X \} the set contaning only the empty subset of XX and all of XX itself;

    this is called the codiscrete topology on XX, it is the coarsest topology (def. ) on XX

    we write CoDisc(X)CoDisc(X) for the resulting topological space.


Last revised on May 1, 2024 at 03:33:23. See the history of this page for a list of all contributions to it.