covering space of S^1

Let p: (0,10) --> S^1, p(t) = (cos t, sin t). Show that p is a local homeomorphism, but ((0,10), p) is NOT a covering space of S^1

surjective covering with finite

Let X and Y connected, locally path connected and Hausdorff. let X be compact. Let f: X ---> Y be a local homeomorphism. Prove that f is a surjective covering with finite fibers. Prove: a) Any subspace of a weak Hausdorff space is weak Hausdorff. b)Any open subset U of a compactly generated space X is compactly generated if each point has an open neighborhood in X with closure contained in U. c)Show that a space is Tychonoff iff it can be embedded in a cube. d) There are Tychonoff spaces that are not k-spaces, but every cube is a compact Hausdorff space.

homotopy type

How can I show that the two contactible have same homotopy type?

homotopy type ; partial order = category

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

free group

show that G1*G2 is not abelian and must contain an element of infinite order.