nLab
separable metric space

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

Idea

In topology, a separable metric space is a topological space that is both separable and metrizable.

Properties

Dimension

Proposition

For separable metric spaces, the following notions of dimension all (exist and) coincide and are thus uniformly referred to as the dimension of a separable metric space:

  1. small and large inductive dimension;

  2. covering dimension.

(e.g. Engelking 78, Theorem 1.7.7)

References

  • Ryszard Engelking, Dimension Theory, Mathematical Library 19, North-Holland Publishing/Polish Scientific Publishers 1978 (pdf)

  • Ryszard Engelking, Theory of Dimensions – Finite and Infinite, Sigma Series in Pure Mathematics 10, Helderman 1995 (pdf)

Last revised on March 21, 2021 at 10:51:02. See the history of this page for a list of all contributions to it.