Tychonoff and Hausdorff Spaces

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.