nLab
Möbius strip
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

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
Idea
The Möbius strip is the topological space obtained from the “open strip”, hence the square , by gluing two of its opposite sites but after applying half of a full rotation to one of them.

Realizations
As a quotient of the open strip
As a topological space , the Möbius strip is the quotient topological space obtained from the square $[0,1]^2$ by the equivalence relation which identifies two of the opposite sides, but with opposite orientation

$((x_1,y_1) \sim (x_2,y_2))
\;\;\Leftrightarrow \;\;
\left(
(x_1,y_1) = (x_2, y_2)
\;\text{or}\;
\left(
x_1 = (1-x_2) \in \{0,1\}
\;\text{and}\;
y_1 = (1-y_2)
\right)
\right)$

As the tautological line bundle over $\mathbb{R}P^1$
Regarded a vector bundle over the circle, the Möbius strip is the tautological line bundle over the 1-dimensional real projective space $\mathbb{P}P^1$ .

For more discussion of the topological vector bundle structure see this example and this prop.

Properties
Regarded as a manifold , the Möbius strip is among the simplest examples of a manifold that is not orientable .

Regarded as a real vector bundle over the circle , the Möbius strip is among the simplest examples of a non-trivial vector bundle .

References
Named after August Möbius .

Terry Lawson, Topology: A Geometric Approach , Oxford University Press (2003) (pdf )
Last revised on November 22, 2020 at 19:54:02.
See the history of this page for a list of all contributions to it.