nLab
neighborhood base

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

Let (X,τ)(X, \tau) be a topological space, and xXx \in X a point. A neighborhood base at xx is a collection {U iX} iI\{U_i \subset X\}_{i \in I} of neighbourhoods of xx such that every neighborhood WW of xx (which WLOG we may assume open) contains some U iU_i:

WopenXW{x}(iI(U iW)). \underset{ { W \underset{\text{open}}{\subset} X } \atop { W \supset \{x\} } }{\forall} \left( \underset{i \in I}{\exists} \left( U_i \subset W \right) \right) \,.

References

See also

Revised on May 15, 2017 06:43:57 by Todd Trimble (67.81.95.215)