Originally Posted by

**joeyjoejoe** I'm looking for a way to show that a finite topological space with some prescribed topology is a contractible space. I understand the definition well enough: I need two continuous functions f,g such that their compositions are homotopic to the identity map on each space, where one of these spaces is just a 1-element finite topological space.

So as an example, say:

x = {a,b}

Tx = {empty set, {a}, x}

y = {a}

Ty = {empty set, {a}}

f: X -> Y is continuous

g: Y -> X is continuous

What would f and g be in this case to make the homotopies work? It seems like one of these functions would have to map a in Y to both a and b in X in order to make them homotopic to the identity functions? Am I not seeing something obvious here? Thanks.