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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?

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    Hi Robb. To specify a mapping, you need more than the domain and range, but also an assignment of every point in the domain to a point in the range. That is, you haven't actually specified a map from Room A to Room B!2011-10-28

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