I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So consider the following mappings:
$f: \text{Room} \ A \to \text{Room} \ B$ and $g: \text{Room} \ A \to \text{Room} \ B$
where $f$ maps points in Room A to the 8 corners of Room B depending on some rules. Also $g$ maps points in Room A to the center of the faces of Room B depending on some rules.
Is the relationship between these two mappings basically what a homotopy is? That is, is it a function $h: f \to g$ (a function between functions)? Suppose we had the following: $f: \text{Room} \ A \to \text{Room} \ B$ and $i: \text{Room} \ A \to \text{Room} \ C$ where $i$ maps points in Room A to the floor of Room C depending on some rules.
Could we consider the relationship between $f$ and $i$? Would this be a homotopy?