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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

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.

Solution Preview

For each x in X we denote by H(x) an open set of X, containing x, such that
f[H(x)] is open and f:H(x) --> f[H(x)] is a homeomorphism.
These exist by f being a local homeomorphism.

Note that f is an open map, so that f[X] is (non-empty) open in Y.
Also, X is compact, so f[X] is a compact subset of the Hausdorff space Y,
so f[X] is closed. But in a connected space, the only non-empty closed-and-open set is Y.
So f[X] = Y and ...

Solution Summary

Tychonoff and Hausdorff Spaces are investigated. The solution is detailed and well presented.