nLab
meridian
Contents
Context
Spheres
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

Manifolds and cobordisms
Contents
Idea
Given an n-sphere $S^n$ for $n \,\geq 1\,$ , regarded under its standard unit sphere embedding into Cartesian space

$S^n
\;\subset\;
\mathbb{R}^{n+1} \,\simeq\, \mathbb{R} \times \mathbb{R}^n
\,,$

then a meridian is any geodesic (great half-circle) from the “north pole” $\big(1,\vec 0\big)$ to the “south pole” $\big(-1,\vec 0\big)$ (or more generally between any pair of antipodes ).

Hence the $n$ -sphere may be regarded as the union of

its north pole

its south pole

all meridians.

This decomposition exhibits the $n$ -sphere as the (un-reduced) suspension of the $(n-1)$ -sphere which is its equator .

References
See also:

Last revised on January 5, 2023 at 19:12:58.
See the history of this page for a list of all contributions to it.