Homotopy is an equivalence relation on the set of all continuous maps between spaces

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

this is rather messy, but it is better than nothing. Currently I'm not sure on the phrasing (eg: "$f$ is homotopic to $g$ is an equivalence relation" or "$f$ and $g$ being homotopic defines an equivalence relation"). It needs to be cleaned up, but it'll do for now

Statement

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces. Let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$. Then:

• the relation that relates a continuous map, $f:X\rightarrow Y$ to another continuous map, $g:X\rightarrow Y$, if $f$ and $g$ are homotopic is an equivalence relation. Symbolically
  • Let $C^0(X,Y)$ denote the set of all continuous maps of the form $(:X\rightarrow Y)$
  • we define a relation, $\mathcal{R}\subseteq C^0(X,Y)\times C^0(X,Y)$ given by
    • $\forall f,g\in C^0(X,Y)[(f,g)\in\mathcal{R}\iff[f\simeq g\ (\text{rel }A)]]$ - recall: $f\simeq g\ (\text{rel }A)$ denotes that the maps $f$ and $g$ are homotopic relative to $A$
  • Then we claim $\mathcal{R}$ is an equivalence relation

We will denote this not by $f\mathcal{R}g$ (as is usual with relations) but by:

1. $f\simeq g\ (\text{rel }A)$ or
2. $f\simeq g$ if $A=\emptyset$

Proof

Grade: B

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Check the proof!

To be an equivalence relation we must show:$\newcommand{\homo}[2]{#1\simeq #2\ (\text{rel }A)}$

1. For all $f\in C^0(X,Y)$ that $f\simeq f\ (\text{rel }A)$, symbolically:
   • Reflexive: $\forall f\in C^0(X,Y)[\homo{f}{f}]$
2. if $f\simeq g\ (\text{rel }A)$ then $g\simeq f\ (\text{rel }A)$, symbolically:
   • Symmetric: $\forall f,g\in C^0(X,Y)[\homo{f}{g}\implies\homo{g}{f}]$
3. If $\homo{f}{g}$ and $\homo{g}{h}$ then $\homo{f}{h}$, symbolically:
   • Transitive: $\forall f,g,h\in C^0(X,Y)\left[\big(\homo{f}{g}\wedge\homo{g}{h}\big)\implies\homo{f}{h}\right]$

Where we are given topological spaces, $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$, and also an arbitrary subset of $X$, $A\in\mathcal{P}(X)$

Reflexive property

Let $f\in C^0(X,Y)$ be given. We want to show that $\homo{f}{f}$.

• Define a map, $H:X\times I\rightarrow Y$ by $H:(x,t)\mapsto f(x)$.
  • If we show $H$ is a homotopy $(\text{rel }A)$ we have exhibited a homotopy between $f$ and itself, thus showing that $f$ is homotopic to $f$ $(\text{rel }A)$

Symmetric property

Let $f,g\in C^0(X,Y)$ be given and suppose that $H:\homo{f}{g}$ is also given. We want to show $\homo{g}{f}$

• Define a map, $H':X\times I\rightarrow Y$ as $H':(x,t)\mapsto H(x,1-t)$.
  • If this map is a homotopy $(\text{rel }A)$ whose initial stage is $g$ and whose final stage is $f$ we are done.

Transitive property

Let $f,g,h\in C^0(X,Y)$ be given and suppose that $F:\homo{f}{g}$ and $G:\homo{g}{h}$ are also given. We want to show that $\homo{f}{h}$

• NOTE TO READERS: I've yet to do this page but this is the only non-trivial result, to answer the question:
  • Define a map $H:X\times I\rightarrow Y$ as $H:(x,t)\mapsto\left\{\begin{array}{lr}F(x,2t) & \text{if }t\in[0,\frac{1}{2}]\\ G(x,2t-1)&\text{if }t\in[\frac{1}{2},1]\end{array}\right.$ (the $t=\frac{1}{2}$ case is a deliberate use of notation to draw attention to the fact that the maps are the same there, see pasting lemma) and showing that is a homotopy