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

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

Statement

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

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

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

1. [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] or
2. [ilmath]f\simeq g[/ilmath] if [ilmath]A=\emptyset[/ilmath]

Proof

Grade: B

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Plenty of sources for proof if one gets stuck, however it should be pretty standard