nLab
Euclidean topology
Context
Topology
topology (point-set topology )

see also algebraic topology , functional analysis and homotopy theory

Introduction

Basic concepts
open subset , closed subset , neighbourhood

topological space (see also locale )

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

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

topological vector space , Banach space , Hilbert space

topological group

topological manifold

cell complex , CW-complex

Examples
empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

sphere , ball ,

circle , torus , annulus

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

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

Zariski topology

Cantor space , Mandelbrot space

Peano curve

long line , line with two origins

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements
Theorems
Theorems

Contents
Definition
Properties
Proposition
Two Cartesian space s $\mathbb{R}^k$ and $\mathbb{R}^l$ (with the Euclidean topology) are homeomorphic precisely if $k = l$ .

A proof of this statement was an early success of algebraic topology .

Revised on January 15, 2011 04:56:57
by

Toby Bartels
(98.19.56.183)