I have a problem in algebraic topology:
We defined Homotopy between spaces like this:
Two top. spaces are called homotopic equivalent, if there are two continuous maps $f:X\to Y$ and $g:Y\to X$ s.t. $(f\circ g)(x)\simeq 1_Y$ and $(g\circ f)(x)\simeq 1_X$.
So my question is: How can a top. space $X$ be homotopic to a point (i.e. contractible), if it has more than one point?
because if we take that one point as $Y$ in the definition of homotopic, we can't find $f$ and $g$ like that, because $g$ can have only one value, because it's domain is one point.