Here is my definition for homotopy:
Let $X,Y$ be topological spaces.
Let $f,g:X\rightarrow Y$ be continuous functions.
If there is a function $F:X\times[0,1]\rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$, then $F$ is called a homopoty between $f,g$ and $f,g$ are said to be homotopic.
However my professor defined homotopy in the following way:
Let $X,Y$ be topolgicial spaces and $A\subset X$.
Let $f,g:X\rightarrow Y$ be continuous functions such that $f=g$ on $A$.
If there is a function $F:X\times[0,1]\rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ and $F(a,t)=f(a) (a\in A, t\in [0,1])$ then $F$ is called a homopoty between $f,g$ on $A$ and $f,g$ are said to be homotopic on $A$.
If $A=\emptyset$, we call $f,g$ are homotopic.
I think my professors definition is extremely weird. I have looked up a page at wikipedia about homotopy, but I couldn't see any category about this.
What is this homotopy called generally? (i.e. In my language, homotopy such that $F(a,t)=f(a) (a\in A, t\in [0,1]$))
Hey so I was just on Wikipedia and able to find a section that has what you are asking about http://en.wikipedia.org/wiki/Homotopy#Relative_homotopy