nLab finer topology

Contents

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

(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.

Examples

Example

(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.

References

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