Sometimes a space $X$ is mapped over another space $Y$ by two maps that are different, but that are in some way compatible, in the sense that "one map may be continuously deformed to match the another map"
How to formally describe this property?
Let $X$, $Y$ be two topological spaces, $f$ and $g:X\longrightarrow Y$ two maps. We say that $f$ and $g$ are homotopic maps if there exists $H:X\times [0,1]\longrightarrow Y$ such that $H(x,0)=f(x)$, $H(x,1)=g(x)$, for all $x\in X$.
We denote $f\sim g$ (or $f\sim_H g$ if $H$ is wanted to be explicit). $H$ is called the homotopy (between $f$ and $g$).
It also may happen that we want to describe the previous phenomenon but stresssing at the same time that the image of some subset of $X$ remains the same all over the deformation
Let $A\subset X$, $f$, $g:X\longrightarrow Y$ with $f\vert_A=g\vert_A$. We say that $f$ and $g$ are homotopic maps relative to $A$ if there exists $H:X\times [0,1]\longrightarrow Y$ such that $H(x,0)=f(x)$, $H(x,1)=g(x)$, for all $x\in X$, and $H(a,s)=f(a)=g(a)$, for all $a\in A$, $s\in [0,1]$. It is denoted $f\sim g (A)$.