nLab
topological interval

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

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

Generally, a topological interval is a (bounded) interval in the real line (an open interval (a,b)(a,b) or a closed interval [a,b][a,b] or a half-open interval (a,b](a,b] or [a,b)[a,b)) equipped with the subspace topology of the Euclidean metric topology.

Specifically in the context of topological homotopy theory, the standard topological interval object is the closed interval [0,1][0,1] equipped with the continuous functions

  1. const 0:*[0,1]const_0 \;\colon\; \ast \to [0,1]

  2. const 1:*[0,1]const_1 \;\colon\; \ast \to [0,1]

which include the point space as the two endpoints, respectively.

Together with the unique function [0,1]*[0,1] \to \ast this yields the factorization of the codiagonal on the point space

*:**(const 0,const 1)[0,1]!* \nabla_{\ast} \;\colon\; \ast \sqcup \ast \overset{(const_0,const_1)}{\longrightarrow} [0,1] \overset{\exists!}{\longrightarrow} \ast

which exhibits an example of an interval object in the general sense of model category theory with respect to the classical model structure on topological spaces.

Induced constructions

Topological cylinder

For XX a topological space, then the product topological space X×[0,1]X \times [0,1] with the topological interval is the standard topological cylinder over XX. Via the above inclusion functions, this inherits a factorization of the codiagonal of XX (the canonical continuous function out of the disjoint union space of XX with itself to XX):

X:XX(id X×const 0,id X×const 1)X×[0,1]overetpr 1X. \nabla_X \;\colon\; X \sqcup X \overset{(id_X \times const_0, id_X \times const_1)}{\longrightarrow} X \times [0,1] \overet{pr_1}{\longrightarrow} X \,.

Accordingly, with respect to the classical model structure on topological spaces this is an example a of cylinder object.

Left homotopy

Given X,YX,Y two topological spaces and f,g:XYf,g \;\colon\; X \longrightarrow Y two continuous functions,then a left homotopy

η:f Lg \eta \colon f \,\Rightarrow_L\, g

is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the product topological space of XX with the topological interval, such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.

(graphics grabbed from J. Tauber here)

Path space

For XX a topological space, then its path space is the mapping space X [0,1]X^{[0,1]}, out of the topological interval into XX, i.e. the set of continuous function γ:[0,1]X\gamma \;\colon\; [0,1] \to X equipped with the compact-open topology.

The two endpoint inclusions *:[0,1]\ast \colon [0,1] and the unique projection [0,1]*[0,1] \to \ast induce continuous functions

XX [0,1]X (const 0,const 1)X×X X \overset{}{\longrightarrow} X^{[0,1]} \overset{X^{(const_0,const_1)}}{\longrightarrow} X \times X

(inclusion of constant paths and endpoint evaluation of paths).

Properties

Freyd’s characterization

The topological interval [0,1][0, 1] may be characterized by a coalgebraic definition first identified by Freyd:

Let Top *,*Top_{\ast, \ast} be the category of topological spaces XX equipped with a pair x 0,x 1x_0, x_1 of distinct points, for example I=([0,1];0,1)I = ([0, 1]; 0, 1). Let F:Top *,*Top *,*F \colon Top_{\ast, \ast} \to Top_{\ast, \ast} be the functor defined on objects by

F(X;x 0,x 1)=(XX,y 0,y 1), F(X; x_0, x_1) = (X \vee X, y_0, y_1) \,,

where XXX \vee X denotes the quotient space of the disjoint union space XXX \sqcup X formed by identifying the element x 1x_1 of the first copy of XX with x 0x_0 of the second copy of XX, where y 0y_0 is identified with x 0x_0 of the first copy, and y 1y_1 is identified with x 1x_1 of the second copy.

For I=([0,1];0,1)I = ([0, 1]; 0, 1) there is an evident identification F(I)=([0,2];0,2)F(I) = ([0, 2]; 0, 2), and moreover there is an FF-coalgebra structure IF(I)I \to F(I) given by “multiplication by 22”.

Theorem

(Freyd)

The topological interval I=([0,1],0,1)I = ([0, 1], 0, 1) is the terminal F-coalgebra.

For more information, see coalgebra of the real interval, which shows in particular how the interval structure of [0,1][0, 1] may be defined by coinduction.

Last revised on July 2, 2017 at 07:54:08. See the history of this page for a list of all contributions to it.