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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Definition
Let $(X, \tau)$ be a topological space , and $x \in X$ a point. A neighborhood base at $x$ is a collection $\{U_i \subset X\}_{i \in I}$ of neighbourhoods of $x$ such that every neighborhood $W$ of $x$ (which WLOG we may assume open) contains some $U_i$ :

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

References
See also

Revised on May 15, 2017 06:43:57
by

Todd Trimble
(67.81.95.215)