nLab
finer topology

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 \in 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 2\tau_2

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

Examples

Example

(discrete and co-discrete topology)

Let SS 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. τ∶−P(S)\tau \coloneq P(S) the set of all open subsets;

    this is called the discrete topology on SS, it is the finest topology (def. 1) on XX,

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

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

    this is called the codiscrete topology on SS, it is the coarsest topology (def. 1) on XX

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

References

Revised on July 10, 2017 23:19:18 by Arya? (2602:30a:2c83:4470:20e1:5992:1090:6849)