sierpinski space is contractible

Let X be Sierpinski space: X={x,y} with topology {X,empty set, {x}} . prove that X is contractible.

"having the same homotopy type" is an equivalence relation

Prove that "having the same homotopy type" is an equivalence relation on the set of topological spaces.

partially-ordered set

Show that a poset (partially-ordered set) is the same thing as a category in which all Hom-sets have at most one element.

partially ordered by inclusion

Let X be any vector space over the field F, let L be a linearly independent subset of X, and A be the set of linearly independent subsets of X containing L. Then A is partially ordered by inclusion - why does it follow?

homology group

Determine the structure of the homology group H_n(X), n >= 0, if X is (a) the set of rational numbers with their usual topology; (b) a countable, discrete set.